Questions tagged [l-functions]
Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.
473 questions
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Problem understanding the proof of $L(s,\operatorname{Ind}_N^H(\eta))=L(s,\eta)$ (Edit)
(I majorly editted the question to improve clarity)
I'm following Jurgen Neukirch's proof of $L(s,\operatorname{Ind}_H^G(\eta))=L(s,\eta)$ and I am having trouble understanding one key step.
The set ...
2
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1
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Zero density estimates for zeta translated to Dirichlet $L$-functions
In this recent paper of Bellotti the author gives a new zero-density estimate for $\zeta(s)$ close to the boundary of the Vinogradov-Korobov region. It is natural to ask if this generalises to ...
user574181
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Are abelian varieties over $\mathbb{F}_q[t]$ which have the same $L$-function isogenous?
Faltings' isogeny theorem implies that two abelian varieties over a number field $K$ which have the same characteristic polynomial of Frobenius at almost all primes are isogenous. If $K = \mathbb{Q}$ ...
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Linear relations between critical $L$-values of cusp forms
Let $k\geq 3$ be an integer, fix a level $N$, consider the critical $L$-values of all cusp forms in this level and weight with algebraic Fourier coefficients:
$$\mathbb{L}_{N,k}:= \{(2\pi)^{k-1-s} L(f,...
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Why Should One Expect Rankin-Selberg Convolutions to Have Certain Conductors, Gamma Factors, etc
Let $L(s,f)$ and $L(s,g)$ be two $L$-functions of degrees $d_{f}$ and $d_{g}$ with local roots $\alpha_{j}(p)$ and $\beta_{\ell}(p)$ at $p$ and $\kappa_{j}$ and local roots $\nu_{\ell}$ at $\infty$. ...
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$L$-function form of Tate conjecture for divisors on abelian varieties
Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
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Size of coefficients of $L$ functions in Selberg $S$ class
Let $F(s) = \sum_n a_n n^{-s}$ be an $L$ function of degree $d$ in the sense of Selberg.
What do we know about the $\sum_n |a_n|$, conjecturally or unconditionally?
I would guess (based on 'vibes' - I ...
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1
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Approximate functional equation for derivatives of Dirichlet $L$-function
I am looking for a reference in the literature which gives the approximate functional equation for $L^{(n)}(s,\chi)$ for a Dirichlet $L$-function.
I know for $\zeta^{(n)}(s)$ we have such results of ...
user566642
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2
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Fast converging series for $L(2,(\frac{d}{\cdot}))$
Dirichlet's $L$-function plays a central role in analytic number theory. For any integer $d\equiv0,1\pmod4$, let
$$L_d(2):=L\left(2,\left(\frac{d}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac dk)...
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How to use Kuznetsov formula?
I read some paper and get that Kuznetsov formula can transform $\sum_c \frac{S(m, n ; c)}{c} g\left(\frac{4 \pi \sqrt{m n}}{c}\right)$ into some information from automorphic form. I wonder if we do ...
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Non vanishing of modular L functions on real line
In M. Ram Murty's paper "Oscillations of Fourier coefficients of modular forms", Math. Ann. 262, 431-446 (1983), MR696516, Zbl 0489.10020 (an offprint can be found here), I see a conjecture (...
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Bounds for logarithmic derivatives of automorphic L-functions
Consider the following results about the logarithmic derivative of the Riemann zeta function:
(Montogomery--Vaughan Lemma 12.2) For each real number $T \geq 2$, there is a $T_1$ with $T \leq T_1 \leq ...
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Ramanujan-Petersson conjecture over SL(3, Z)
Recently, I am reading the paper *"Twisted moments of L-functions and spectral reciprocity" by Blomer, Valentin and
Khan, Rizwanur in 2019 (arXiv, DOI).
I wonder what is the Ramanujan-...
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Functional equation of L-function with associated squared representation function by quadratic form
Let $r_Q(n)=\#\{x\in \mathbb{Z}^m|Q(x)=n\}$ be the number of representations of an integer $n$ by a definite quadratic form Q, and define the L-function $$\zeta_Q(s)=\sum_{n=1}^\infty r_Q(n)n^{-s}$$ ...
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The Hurwitz zeta function on the line $\mathrm{Re}(s) = 1$
For $\mathrm{Re}(s) > 1$ and $0 < a \leqslant 1$, let $\zeta(s,a) = \sum_{n \geqslant 0} 1/(n+a)^s$ denote the Hurwitz zeta function. As a function of $s$, $\zeta(s,a)$ has a meromorphic ...
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$L$-functions and Stirling's formula
Define a Dirichlet series $\zeta_X^D(s)$ by the relation
$$ \left(-\log \zeta_X^D(s) \right)' \; \zeta(s) = \sum_{n\geq 1} \frac{\log\left( n!_X^D/(n-1)!_X^D\right)}{n^s}, $$
where $n!_X^D = |D/v_n(X,...
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Is there a connection between Selberg's conjecture and the Burgess Bound / The Weyl Bound?
This is a repost from my question on Math Stack(https://math.stackexchange.com/questions/5017788/is-there-a-connection-between-selbergs-conjecture-and-the-burgess-bound-the-w?noredirect=1#...
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Relationship Between Zeros of Induced Dirichlet Characters
I was reading the paper "Counting Zeros of Dirichlet L-functions" by Bennett et al and came across the following statement at the end of page 1:
Let $Z(\chi) $:= $\{\rho \in \mathbb{C} : 0 &...
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Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
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Distinguishedness of discrete series induction
Let $D_k$ denote a discrete series representation of $\text{GL}_2(\mathbb{R})$ of weight $k\geq 2$. Consider the parabolically induced representation $D_k \times D_k$, which is a representation of $\...
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The conductor of twists of $GL(d)$ $L$-function
It is from ANT of Iwaniec and Kowalski that the conductor of $L$-function is defined that:
An integer $q(f) \geqslant 1$, called the conductor of $L(f, s)$, such that $\alpha_i(p) \neq 0$ for $p \nmid ...
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Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
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Factorization of global Waldspurger's integrals and connection to central L-values
Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$.
For $f_1 \in \pi$, $f_2 \...
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Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
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A question on vanishing of Godement–Jacquet-like zeta integral
$\DeclareMathOperator\GL{GL}$The classical Godement–Jacquet zeta integral is of this form:
$f$ is a matrix coefficient of a cuspidal automorphic representation of $\GL_n(\mathbb{A}_\mathbb{Q})$, and $\...
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Which L-functions are not known to be automorphic for $\mathrm{GL}_n$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I would like to compile a list of primitive L-functions which satisfy
the usual axioms (Dirichlet series with an Euler product,
and a ...
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To what extent are Langlands conjectural global L-functions unique?
Question
To what extent do the properties that are conjectured of L-functions determine them?
Explanation
Following Shahidi: So Langlands defines local L functions associated to unramified ...
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Generalizations of Hamburger's Theorem
(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).
An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
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Holomorphy of integral representation
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
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Evidence for the equivariant BSD conjecture with higher multiplicity
Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho$ be an irreducible Artin representation. Let $K_\rho/\mathbb{Q}$ be the smallest Galois extension such that $\rho$ factors through $\mathrm{Gal}(...
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Integral representations of Dirichlet L-function of quantum modular forms
It's known that holomorphic Eisenstein series for odd weight vanish in their lattice sum representation. However, its definition can be extended to allow for odd integers via its $q$-expansion
$$G_k=2\...
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Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
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The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
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A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
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Bounds for Dirichlet L-functions
Let $L$ denote a Dirichlet L-function attached to the primitive character $\chi$. What are the best known bounds for $L(\sigma+it, \chi)$?
PS: For $L=\zeta$ and $0\leq\sigma\leq 1$, i'm aware of a ...
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The product of Petersson norm of Hecke eigenforms
Let $f_1,\dots,f_k$ be the normalized Hecke eigenforms in $S_{12k}(\operatorname{SL}_2(\mathbb{Z}))$. Do we have asymptotic formula for the quantity $\prod_{i=1}^k \langle f_i,f_i \rangle_{\...
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What are $L$-functions?
I am coming at this question from the point of view of someone who is working in arithmetic geometry around the Langlands program.
We have $L$-functions associated to many different structures that we ...
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Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$
Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\...
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Arithmetic interest of the Goss zeta function
I'm someone with more of a number fields background who recently started working on a project more in the function fields setting. I was reading Goss's book (Basic structures of function field ...
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Symmetric square $L$-functions over imaginary quadratic field
Let $F = \mathbb{Q}(\sqrt{-d})$ with class number $h_F = 1$, and $\Gamma = \mathrm{PSL}_2(\mathfrak{O}_F)$. Let $f$ be a Maass cusp form in the $L^2$-cuspidal spectrum of the Laplace operator $\...
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Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
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Classification of L functions and Dirichlet series by poles
I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions.
Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
user523765
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2
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Conditional convergence of exponential sums related to a Hecke modular form
Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
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A question about the setup of zero density estimates for Dirichlet $L$-functions
For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
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Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
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The logarithmic derivative of a twisted L-function?
Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have
$$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$
(I ...
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2
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Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?
Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere:
Let $E$ be an elliptic curve, and $f(q)$ its associated modular ...
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0
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170
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Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
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1
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341
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'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 315
1
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0
answers
270
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Behavior of Dirichlet L-functions at the edge of the critical strip
Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
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