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      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.

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      3answers
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      Are L-functions uniquely determined by their values at negative integers?

      Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that the corresponding L-function $$L_{\{a_n\}...
      -2
      votes
      0answers
      110 views

      Does the strong multiplicity one theorem imply the isomorphy of these two automorphism groups?

      Defining the notion of "Galois class of L-functions" as a set of automorphic L-functions belonging to the Selberg class closed under the usual product and the Rankin-Selberg convolution and containing ...
      4
      votes
      0answers
      132 views

      Change of variables for $p$-adic integral

      Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
      3
      votes
      1answer
      127 views

      Explicit formula: explicit work with general smoothing?

      The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
      3
      votes
      0answers
      74 views

      Similarity between two $L$-functions (Hasse-Weil $L$-function of twisted ellptic curve and Dirichlet $L$-function)

      Let $E$ be an elliptic curve over $\mathbb Q$ with conductor $N$ and $E_d$ be its twisted curve by $d$, where $d$ is a fundamental discriminant with $(d,N)=1$. Let $\chi_d$ be a Dirichlet character ...
      2
      votes
      1answer
      95 views

      Complex L-functions for Hermitian modular forms?

      Fix an imaginary quadratic field $K$, and let $\mathcal{O}_K$ be its ring of integers. A Hermitian modular form of genus 1 (i.e., an automorphic form on $GU(1,1)$) of weight $(k_1,k_2)$ on a ...
      4
      votes
      0answers
      106 views

      Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?

      Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
      2
      votes
      0answers
      83 views

      Adelic Mellin transform with nontrivial character

      This is a short question with a lot of setup. I apologize in advance. In Dan Bump's "Automorphic Forms and Representations," he constructs the L-function of a modular form via an "adelic Mellin ...
      2
      votes
      0answers
      104 views

      Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$

      This is a very short question. Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$. In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
      10
      votes
      1answer
      569 views

      Least quadratic residue under GRH: an explicit bound

      Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
      4
      votes
      2answers
      140 views

      Real non trivial zeros of Dirichlet L-functions

      When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
      4
      votes
      0answers
      125 views

      Second derivative at 1 of L function of elliptic curve

      Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that $$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$ where $\gamma$ is Euler's ...
      5
      votes
      2answers
      171 views

      Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

      Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
      7
      votes
      0answers
      227 views

      A mysterious number related to Hasse-Weil L-function of elliptic curve

      Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
      3
      votes
      2answers
      78 views

      Why we use Caputo fractional derivative in application?

      I'm working on some papers which use Caputo fractional evolution equation as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{...

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