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, ...

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0answers
74 views

On the computation of Asai L-function

I want so compute some simple twisted Asai L-function. Let $E/F$ be a quadratic extemsion of number fields and $v$ a finite place of $F$. Let $\chi$ be a unitary automorphic character of ...
1
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1answer
92 views

On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$. Let $\chi_v$ be a unramified unitary character of $F_v$. Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...
4
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1answer
110 views

Characterizing the newforms s.t. the associated symmetric square $L$-function has a pole

I have a straightforward question. Let $f$ be a holomorphic cusp form of weight $k$, level $N$, and nebentypus $\chi$ that is new in the sense of Atkin-Lehner theory. Write its Fourier expansion at ...
2
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1answer
233 views

Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture? Is there some book (or expository article) about this conjecture? Is there any connection between this conjecture and ...
5
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1answer
224 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
8
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0answers
153 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
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2answers
395 views

GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome. In light the local Langlands correspondence, what ...
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1answer
240 views

Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$. Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$. ...
5
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1answer
206 views

absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...
2
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2answers
287 views

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
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1answer
338 views

Is SOC known to imply the Grand Riemann Hypothesis? [closed]

I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...
2
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1answer
97 views

Are there L-functions of degree 1 that aren't Hecke L-functions?

I don't know of any examples and I don't know of any results which prohibit them
2
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1answer
230 views

Automorphic L-functions over $GL_n( \mathbb{Q} )$

A paper by Kaczorowski & Perelli arXiv:1207.2312 dealing with the elements of the Selber class with degree two suggests that $S_2$ coincides with the automorphic l-functions over $GL_2( \mathbb{Q} ...
3
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0answers
79 views

Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2). Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?
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1answer
662 views

Main conjecture for elliptic curves

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...
3
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2answers
136 views

Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...
5
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1answer
228 views

p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?

Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating ...
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0answers
147 views

Is there a “natural” reformulation of Hodge conjecture in terms of L-functions?

I just glanced at the Wikipedia article about the Hodge conjecture, and a (probably very naive, due to my huge lack of knowledge of the subject) question just came to my mind: can one associate ...
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0answers
73 views

Has universality been definitely established for the whole Selberg class?

I juste googled to get some insight about universality for l-functions belonging to the Selberg class, but it seems that the proof requires the validity of the prime number theorem for the considered ...
3
votes
2answers
159 views

l-functions of calabi-yau varieties

This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
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0answers
248 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
6
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1answer
288 views

Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean: If $\phi_n$'s are ...
8
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0answers
140 views

Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...
5
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0answers
216 views

A sum over zeros of L-functions in the paper “Chebychev's Bias”

Let $\varepsilon>0$ be small and \begin{align*} \widetilde{F}_{\varepsilon}(\xi):=\frac{4}{\varepsilon}\sum_{0<\gamma\leq \varepsilon^{-2}}\frac{\sin(\gamma \xi)\sin \frac{\gamma ...
31
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0answers
723 views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
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0answers
203 views

Automorphisms of an L-function

Throughout this question, the term "L-function" will denote any element of the Selberg class. Following Strong automorphisms of the Selberg class, I define the group of automorphisms of an L-function ...
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0answers
99 views

A $GL_1$ Voronoi formula

I want a functional equation for the function defined by the Dirichlet series, $$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$ which sends $s$ to $1-s$ and preferably sends $a$ to ...
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0answers
50 views

Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
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1answer
286 views

Order of some $L$-function at $s=0$

Sorry, I asked this two days ago, but this time I modified it to be easily read and added more specific explanation. I hope to get your illuminating comment on whether my approach is right. I am ...
7
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0answers
510 views

Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields? More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, ...
2
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0answers
213 views

Convergence of certain L-series

Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values. Suppose further that, $ L(s)=\sum_{n} \frac{a_{n}}{n^s} $ may be continued analytically to the left of $s=1$ a bit ...
4
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1answer
223 views

on the conductor of a scheme over $\mathbb{Z}$

Let $X$ be a regular, projective flat scheme over $Spec(\mathbb{Z})$, of relative dimension $d$, and look at the $L$-function $L(X, s)$, that is, the Hasse-Weil zeta function completed by the Gamma ...
2
votes
1answer
184 views

The effect of base change on the L-function of GL(2)?

Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi_1, \chi_2)$ of $GL_F(2)$ especially where ...
2
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2answers
564 views

L-functions and algebraic geometry

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...
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0answers
97 views

reference needed: Dirichlet L-functions

Where is the logarithmic derivative at $s=0$ of the L-function of a Dirichlet character computed? Many people cite the Hurwitz formula but I was unable to find a suitable reference.
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1answer
111 views

degree of an isobaric sum

I'm trying to understand a few things about automorphic L-functions. In page 5 of http://arxiv.org/pdf/1401.0390.pdf, the author mentions the isobaric sum decomposition ...
4
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1answer
351 views

subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated. ...
8
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1answer
383 views

Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...
3
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0answers
133 views

Are quantities involved in Generalized Ramanujan Conjecture eigenvalues of some unitary operator?

If I'm not mistaken, every automorphic L-function $L(s,\pi)$ verifies $\displaystyle{L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})}$ where ...
2
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1answer
220 views

Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity. My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...
2
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0answers
239 views

Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...
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1answer
1k views

derivatives of Artin L-functions

This is a vague question: I'm sorry for that. Let's start with $\chi$ a (primitive odd) Dirichlet character modulo $n$ and look at the corresponding L-function $$ L(s, \chi)=\sum ...
2
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0answers
178 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
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1answer
171 views

Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?

Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large. Then $$L(-1+it,f)\ll_f \log^c q(f)t$$ holds, for some ...
7
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0answers
1k views

What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them? Edit: Further cases of "fractional motives" as ...
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0answers
224 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
26
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13answers
4k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
18
votes
4answers
2k views

Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions? What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...
3
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1answer
310 views

On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
1
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1answer
117 views

On properties of coefficients of Selberg Class L-function

The coefficient of Selberg Class L-function satisfy: $a_n <M_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a_n$ are multiplicative. So I would like to know if it can be shown that ...