Questions tagged [l-functions]

L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.

110 questions
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Relationship between $\theta$ functions and number fields.

I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to. Firstly, I know that $\theta$-functions arise as sections of ...
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Test vector for local zeta integral with ramified character

Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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Langlands L functions for groups over finite fields.

In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
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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. (1) 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 ...
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Determining elliptic curve analytic rank even/odd

For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if ...
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Permutation group of Satake parameters

Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $G_{p}$ of the ...
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Evaluating Dirichlet $L$-functions at $s=1$

I'm trying to find references on general methods for evaluating Dirichlet $L$-functions at $s=1$, but it's proving a little harder to google than I'd hoped. Specifically I'm looking for any books or ...
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Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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Simplest nontrivial example of an L-function yielding information about a Diophantine equation

I got excited while reading Langlands' essay, REPRESENTATION THEORY: ITS RISE AND ITS ROLE IN NUMBER THEORY, because he appears to provide concrete motivation for the study of L-functions: We have ...
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Artin $L$-functions and abelianization

Let $L/K$ be a finite Galois extension of global fields, with $G=\mathrm{Gal}(L/K)$. Let $H \leq G$ and $\chi:H \to \mathbb{C}$ a non-trivial irreducible character of $H$. Then we can define the (...
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Understanding a result of Serre about zeros of $x^3 - x - 1$ in $\mathbb{F}_p$

I'm trying to understand a result of Serre which relates the number of zeros in the finite field $\mathbb{F}_p$ of $f(x) = x^3 - x - 1$ to a modular form. The result can be found in the section 5.2 of ...
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Reference for $L$-functions of curves

I am looking for a reference that explains as easily and as completely as possible how the $L-$function of a curve $C$ (non-singular, projective, geometrically irreducible, defined over $\mathbb{Q}$, ...
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Washington's Formula for $p$-adic $L$ function

I am reading through Washington's construction of $p$-adic $L$ functions in chapter $5$ of his book Cyclotomic Fields. Instead of $p$-adically interpolating as Kubota and Leopoldt originally did, ...
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Is the Generalized Riemann Hypothesis thought to be true?

The Riemann Zeta Function is the analytic continuation of the following function: $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$ The Riemann Hypothesis states that the zeros of this in the critical ...
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