28 votes
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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? No. The rescaled Riemann zeta function $$ \zeta(2s) = \sum_{m=1}^\infty \frac{1}{m^{2s}} = \sum_{n=1}^\infty \frac{a_n}{n^s}, ...
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  • 1,926
22 votes
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Local factors of Hasse-Weil zeta function - what do they have in common?

This is an elaboration on ACL's answer, way too long for a comment, which highlights a technical ingredient (well-known to all experts) that underlies the precise sense in which the $\ell$-adic etale ...
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  • 1,573
22 votes
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Least quadratic residue under GRH: an explicit bound

See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $\chi$ is a primitive quadratic character (as the title suggests) then ...
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  • 42.4k
22 votes
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The modularity theorem as a special case of the Bloch-Kato conjecture

That is not what the link says. To quote (emphasis mine): ... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic ...
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20 votes
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Relation between Schanuel's theorem and class number equation

A spirit. A general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, via Tauberian arguments: the rightmost pole (resp. residue)...
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19 votes
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How do functional equations for zeta functions arise from the structure of a homology group?

There are examples that don't come from number theory, although it's not much simpler. Specifically, the Lefschetz zeta function. Let $X$ be a compact manifold of dimension $d$ and let $f: X\to X$ be ...
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  • 116k
17 votes

Underlying idea for (automorphic) L-function?

This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on ...
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  • 17k
16 votes
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Primer on Eisenstein series

I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), ...
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  • 21.3k
16 votes
Accepted

Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

A result of Sarnak and Zaharescu, stated in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong ...
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  • 91k
15 votes

"Gross-Zagier" formulae outside of number theory

I think you know all this, but nevertheless... These two formulas are arguably incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (...
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  • 9,500
15 votes
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Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?

Good question. I don't understand fully what's happening, but here is an idea. Let $f=\sum a_n q^n$ be a modular form of weight $k+1/2$, nebentypus $\chi$. Assuming $k \geq 1$, the Shimura ...
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  • 25k
15 votes
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Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$

The right person to answer this question is probably Dinakar Ramakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}...
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15 votes
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Euler factors of L-function at bad primes

1) True. 2) True. 3) True. 4) They are equal in the case of "tame ramification", e.g. if the degree of $K(\mu_p)$ over $K$, $K$ the coefficient field of the motive, is greater than $d$. One just ...
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  • 116k
14 votes

Tate's thesis for Artin L-functions

I think there is a misunderstanding on your side. Tate's thesis is not about special Artin $L$-functions, but about special automorphic $L$-functions. It is a reformulation of Hecke's ideas in a ...
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  • 85.6k
14 votes
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Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?

The de Bruijn-Newman constant is nonnegative, as proved in this brand new preprint by Rodgers and Tao. It is also conjectured, but not proven yet, that $M(x)=O(\sqrt{x})$ is false, in which case it is ...
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  • 85.6k
14 votes
Accepted

BSD conjecture for rank 1 elliptic curves

The following theorem is due to Chris Skinner, in this 2014 paper. Let E/Q be an elliptic curve such that rank E(Q) = 1 and the Tate-Shafarevich group Sha(E / Q) is finite, and some other ...
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14 votes

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

Your restriction that all $a_n$ are integers is too restrictive to define $L$-functions in general (even most Dirichlet $L$-functions are not like that), and you left out the Euler product. Anyway, ...
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  • 41.8k
14 votes
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Hadamard factorization of L-functions

I believe you are correct and $b$ is zero, although I find it inexplicable why this is not better known (certainly I didn't know it before). Let's stick to a primitive Dirichlet character $\mod q$, ...
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  • 42.4k
13 votes

What kind of non-cuspidal automorphic representation are not isobaric sums?

EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not ...
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13 votes

How does Riemann hypothesis implies estimates?

We have that \[-\frac{L'}{L}(s,\mathrm{sym}^2 f) = \sum_{n = 1}^{\infty} \frac{\Lambda_{\mathrm{sym}^2 f}(n)}{n^s},\] where $\Lambda_{\mathrm{sym}^2 f}(n)$ is equal to $\lambda_f(p^2) \log p$ if $n = ...
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13 votes

Some questions on the $p$-adic properties of special $L$-values

1) What generalizations of the Kummer congruences are known? This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that This is probably the same ...
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  • 9,500
12 votes
Accepted

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

I am not sure exactly of what is asked in this question, but I wish to correct what has been said in an answer because it is a little bit outdated. (Edited after Kevin's comment) For $n=2$, Artin's ...
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  • 25k
12 votes

First formulation of the Dedekind and Hasse-Weil conjectures

Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser ...
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12 votes

Intuition for Zagier's theorem for $\zeta_K(2)$

There is a K-theory reformulation of Zagier's theorem which explains both the appearance of the integrals $A(x)$ as well as the appearance of hyperbolic geometry. I will try to give a sketch what is ...
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11 votes

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

Here is an elementary proof, the basic idea for which is in Selberg's 1949 paper "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression" (Ann. Math., vol 2, 1949, pp. ...
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11 votes
Accepted

Subconvexity bounds and zero-free regions

Let's just discuss the $t$-aspect, i.e. bounds for the zeta function and its zeroes. Let $T$ be a large ordinate, and let $H$ be a medium-sized quantity (much larger than $1$, but much less than $T$)....
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  • 91k
11 votes
Accepted

On the consistency of the definition of the conductor for automorphic forms

These definitions are consistent, though it's not immediate. The conductor quantifies the extent to which $\pi$ is ramified. As an aside, I prefer to write $c(\pi)$ for the conductor exponent of $\pi$,...
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11 votes
Accepted

"Extra Euler factors" in one definition of the L-function of a twist of a modular form

Let me translate this into a problem purely about automorphic forms: Take a newform $f \in \mathcal{S}_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a ...
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11 votes

$|L'(1,\chi)/L(1,\chi)|$

You may use the local method of Landau with some bounds for L(s,chi) (expressing L'/L in terms of the local zeros, the approximation being controlled by an upper bound for |L(s,chi)| in a slightly ...
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