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Questions tagged [l-functions]

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

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21 views

Estimation of certain Euler product using a symmetric square $L$-function

Let $$ L(s)=\sum_{n=1}^{\infty}a(n)n^{-s} $$ be a modular $L$-function of conductor $N$, and let $$ F_d(s)=\sum_{n=1}^{\infty} \frac{a(d_0 n^2)}{(d_0 n^2)^s} \prod_{p|4Nnd}\left(1+\frac{1}{p}\right)^...
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Absolute value of roots of bad Euler factors.

Suppose $V$ is a smooth projective variety over number field $\mathbb{Q}$. For every prime integer $p$ and positive integer $j$, let \begin{equation} Z_p(H^j(V), T):=\det(1-{\rm Frob}_p\cdot T|H^j_{et}...
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68 views

Topics for Masters thesis around Birch-Swinnerton-Dyer conjecture [closed]

I am currently looking for a Masters thesis subject in number theory. My favourite subjects are algebraic number theory and cohomologies (I only studied De Rham cohomology). I've been lately reading ...
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36 views

Is the reciprocal of a Satake parameter a Satake parameter?

Let $F$ be an automorphic L-function for $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. Is it known whether for all but finitely many primes $p$ the set of Satake parameters of $F$ at $p$ is ...
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24 views

Do you know about the textbook of Selberg class of Dirichlet series?

I have read the Atle Selberg's thesis named "Old and new conjectures and results about a class of Dirichlet series". At the end of this thesis, he wrote "A more complete account with proofs is ...
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24 views

Probability for an L-function to be RS-primitive

Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
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66 views

Minimum and maximum of a partial Euler product?

Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be $$ \beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}; $$ which for ...
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39 views

$p$- adic $L$- Function of a Dirichlet character at positive integers.

Let $p$ be a prime number and $d$ a natural number with $p\nmid d$. Furthermore let $\chi$ be a Dirichlet character with conductor $d$ and $G=Gal(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$. In the ...
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11 views

Derivative of dirchlet L function at s=1

I'm trying to derive the chowla-selberg formula and it involves the derivative of the dirchlet L function at s=1. I've tried to manipulate the series in several ways, but there's only one that seems ...
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108 views

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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1answer
50 views

Gamma Function Integral Identity

On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function: $$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
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22 views

Expression of the dirichlet coefficient of an L-series in terms of the Satake parameters

Last year, I had found a pdf where the expression of the Dirichlet coefficient $\lambda_{\pi}(p^{\nu})$ in terms of the Satake parameters $\alpha_{p,i}(\pi)$ was given. Unfortunately I don't remember ...
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1answer
61 views

Asymptotics for average of Fourier coefficients of cusp form

Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function $$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$ of a weight $k$ ...
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17 views

The argument of L-functions

For Riemann Zeta function, we can define $\pi S(t)=\arg\zeta(\frac{1}{2}+it)$ where the argument is the variation from $+\infty+it$ to $\frac{1}{2}+it$. I read few results related to $S(t)$ such as $S(...
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51 views

Proof of Chebotarev Density Theorem without Artin Reciprocity

I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity. As I understand it, the main ...
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37 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
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42 views

Need reference for proof of functional equation for generalized L-functions

When proving functional equation for Riemann zeta function one starts at the definition of gamma function $$\Gamma(s) = \int_0^{\infty} x^{s-1} e^x\mathrm dx\tag1$$ After a few steps we arrive at $$ ...
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78 views

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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44 views

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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35 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. (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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116 views

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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29 views

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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132 views

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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46 views

Series of reciprocal of integers

This is a question I asked myself today... $ $ Do you know if it is possible to build a strictly-increasing sequence $(u_n)_{n\in\mathbb{N}^\star}$ of positive integers such that $\displaystyle\sum_{...
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30 views

Is $ \pi\mapsto(s\mapsto L(s,\pi)) $ bijective?

Let $ \pi $ be an automorphic representation of $ \operatorname{GL_{n}}(\mathbb{A}_{\mathbb{Q}}) $ and $ L(s,\pi) $ the associated L-function. Is the map $ \pi\mapsto L(s,\pi) $ bijective ? In ...
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self-intertwining operators of an automorphic representation

I stumbled a few days ago on the definition of an intertwining operator in https://www.encyclopediaofmath.org/index.php/Intertwining_operator Considering the case where $ \pi_{1}=\pi_{2}=\pi$, $ ...
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1answer
23 views

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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77 views

Is the conductor of an L-function F the absolute value of the discriminant ofsome number field related to F?

In the theory of automorphic forms, ramified primes of an L-function divide the so-called conductor thereof. On the other hand, one can define for a number field $ K $ an integral invariant $ \...
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57 views

Uniqueness of $L$-series of cusp forms

For a cusp form $f$, one gets an $L$-series by taking the Mellin transform as we have $$ \tilde{f}(s) = (2\pi)^{-s} \Gamma(s) L(s,f). $$ My question is: is this operation injective? It seems to me ...
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100 views

Convexity Bound of Rankin-Selberg L-Function

Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the ...
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1answer
108 views

Rankin-Selberg convolution : normalization issues

I was told several times on MO that if $ F : s\mapsto\sum_{n>0}\frac{a_{n}}{n^{s}}$ and $ G : s\mapsto\sum_{n>0}\frac{b_{n}}{n^{s}} $ for $ \Re(s)>1 $ are L-functions, then provided the ...
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53 views

The $\heartsuit$ operator on $\mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$

In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(...
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388 views

Roadmap to Iwasawa Theory

I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good ...
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86 views

Upper bound on zero multiplicity of Dirichlet $L$-functions

I was wondering whether there is (a known) upper bound of the order of the non-trivial zeros of Dirichlet $L$-functions. For a zero $s$ of the Riemann zeta function we have the estimate $C\cdot\log(\...
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96 views

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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43 views

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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Why does the Dedekind zeta function of a number field have a pole at $s=1$?

The analytic class number formula tells us that the Dedekind zeta function $\zeta_K$ of a number field $K$ has a pole at $s=1$ with residue $$\frac{2^{r_1}(2\pi)^{r_2}\text{Reg}_Kh_K}{w_K\sqrt{|\...
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94 views

Why does the determinant come in for Artin L-Functions?

Let $L/K$ be a Galois extension of number fields. For $\mathfrak p$ a prime of $K$, unramified in $L$, the Frobenius elements $\sigma_{\mathfrak P}$ for $\mathfrak P \mid \mathfrak p$ are conjugate, ...
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60 views

Trying to understand generic nature of L-functions

I’m not a “professional” mathematician but amateur and curious in maths - more like a hobbyist. So this is a little bit advanced topic for me and am trying to visualize/grasp understanding of how L- ...
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97 views

Analytic continuation of twisted Hecke $L$-function

Let $K$ be a (real quadratic) number field and $\chi$ be a Hecke character on it. We can think about Hecke $L$-function $$ L(\chi, s):= \sum_{0\neq \mathfrak{a}\subseteq \mathcal{O}_{K}} \frac{\chi(\...
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101 views

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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27 views

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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57 views

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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1answer
280 views

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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121 views

How to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$?

I am trying to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$ where $\chi = \mathbb{A}^\times / \mathbb{Q}^\times$ and $\pi$ is a cuspidal representation of $GL_2( \mathbb{A})$ (where $\...
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58 views

Study L-function associated to elliptic curves

I'm studying Iwasawa theory, and now I'm reading Greenberg's article about the application of Iwasawa theory of elliptic curves, but I have some problems on the first chapter, because it's about Hasse ...
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63 views

uniform convergence of $L(s,\chi)$ for $\Re(s) ≥ 1 + \delta$ " due to absolute convergence for $\Re(s)>1$?

On page 6 of this link, lemma 2.4 shows $L(s,\chi)$ is absolutely convergent for $\Re(s)>1$. I understand the proof. However, they also add: "The above proof also shows that for any $\delta > ...
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1answer
98 views

Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $

I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product: $$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 = q \prod_{n=1}^\infty (1 - q^n)...
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1answer
147 views

Generalization of a $\det$ series for $\zeta_{\mathbb{Q}(\zeta_k)}$

With $\mathbb{Q}(\zeta_k)$ a cyclotomic field, $\chi_1,\ldots,\chi_{\phi(k)}$ the Dirichlet characters modulo $k$ and $\tilde{\chi}_1,\ldots,\tilde{\chi}_{\phi(k)}$ the underlying primitive Dirichlet ...
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1answer
218 views

Bounds for Dirichlet L-functions

In the half plane $\sigma$=Re(s) > 1 , one can find bounds for the Riemann zeta function $\zeta$(s) using either its convergent series or product formula.$\,$ From the Dirichlet series we get the ...