# Tag Info

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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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### 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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### 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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### 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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### 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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### 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 ...

### 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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### 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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### 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 ...

### "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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### 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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### 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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### 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 ...

### 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 ...

### 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 ...

### 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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### 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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### 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 $\pi$,...
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 ...