Questions tagged [l-functions]

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

22
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1answer
633 views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field

If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
14
votes
1answer
634 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \...
10
votes
1answer
277 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 ...
10
votes
2answers
140 views
+100

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{|\...
7
votes
1answer
217 views

Sum of reciprocals of primes diverges

I can show that $$\log(\zeta (s)) = \sum _{p\in\Bbb P} \frac{1}{p} + R(s)$$where $$R(s) = \sum _{m\geq 2} \sum_{p\in\Bbb P} \frac{1}{m} \frac{1}{p^{ms}}$$ where $\Bbb P$ is the set of all primes, ...
7
votes
1answer
154 views

Evaluation of $\sum_{m,n=-\infty}^{\infty} (m^2+Pn^2)^{-s}$ where $(m,n)\neq 0$

I was trying to learn about evaluating certain double sums and came across this formula: $$\sum_{\begin{matrix}m,n=-\infty \\ (m,n)\neq (0,0)\end{matrix}}^{\infty} \frac{1}{\left( m^2+Pn^2\right)^{...
6
votes
3answers
205 views

Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{...
6
votes
0answers
240 views

Reference for $L$-functions

What will be a good reference to study $L$-functions for a beginner? Is there any book/lecture note in complex analysis that covers it?
5
votes
1answer
166 views

Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
5
votes
1answer
2k views

what does the “L” in “L-function” stand for?

I haven't been able to find a reference that tells what word (if a word) the L is short for.
5
votes
0answers
320 views

Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form

Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
4
votes
1answer
89 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, ...
4
votes
1answer
70 views

L-function at s=5 with D=-4?

I want to know the value of $L(5,-4)$. Recall that $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right) n^{-s}. $$ I would like a reference with computations of $L(5,D)$, or more generally, of $L(s,...
4
votes
1answer
445 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same $...
4
votes
1answer
606 views

$L$-functions of elliptic curves over $\mathbb{Q}$

How to find out the $P_{v}(E/\mathbb{Q},X)$ theoretically given below in the definition of $L$-functions for elliptic curves over $\mathbb{Q}$ $?$ Please cite some references for the same. For an ...
4
votes
1answer
44 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_{...
4
votes
1answer
314 views

Showing $L(1,\chi)$ is positive given that it's nonzero

Let me first provide context for this question. There is a series of four exercises in Ireland & Rosen's book (in second edition it's exercises 14-17 in chaprer 16), aim of which is (although ...
4
votes
1answer
150 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
4
votes
1answer
86 views

L-series through integrals of rational functions

Recently I stumbled upon this short proof here: $$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$ so: $$\...
4
votes
1answer
72 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 $ \...
4
votes
0answers
101 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 ...
4
votes
1answer
79 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 ...
3
votes
2answers
368 views

What do the zero's of L-functions entail?

I don't know exactly how, but I've read the Riemann Zeta function's nontrivial zero's imply something about an error term for an approximation function thing for the Prime Counting Function. I found ...
3
votes
3answers
610 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
3
votes
1answer
54 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 $k$ ...
3
votes
1answer
141 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 ...
3
votes
1answer
48 views

Eigenforms and L-functions

I'm trying to show the following identity: If $f=\sum_{n\geq 1}a(n)q^n \in S_k$ is a normalized Heckeeigenform, where $k$ can be written as a sum of two even numbers $k= k_1+k_2$, and $L(f,s)=\sum_{n\...
3
votes
1answer
65 views

Determine Fourier coefficients by the values of its L-Series.

suppose I know the values of $\sum_{n=1}^\infty \frac{a_n}{n^k}$ for all $k=1,2,...$. Is there a way/tool to determine the coefficients $a_n$ from this (which might not be unique)? I would appreciate ...
3
votes
2answers
325 views

Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
3
votes
1answer
105 views

$L$-function of character in terms of other character

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$. ...
3
votes
1answer
126 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly appreciated....
3
votes
0answers
89 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 ...
3
votes
0answers
38 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 ...
3
votes
0answers
98 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 ...
3
votes
0answers
66 views

What do $L$-functions of curves over $\mathbb Q$ tell us about the curve

Following up this thread: $L$-function of an elliptic curve and isomorphism class I'd like to ask some more questions for the case of smooth projective curves $C$ over $\mathbb Q$ To be more precise,...
3
votes
0answers
56 views

is this map necessarily a field automorphism?

Let $M$ denote the intersection of the Selberg class and the class of automorphic L-functions and let's define the automorphism group of $M$, denoted by $Aut(M)$, as the group (under composition) of ...
3
votes
1answer
192 views

Dirichlet series 'shifted' by a polynomial

Let $F(x) \in \mathbb{Z}[x]$ and $$ \xi(s) = \sum^\infty_{n=1}g(n)n^{-s} $$ be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series $$ \xi_F(s) = \sum^\...
2
votes
2answers
72 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....
2
votes
2answers
224 views

Functional equation of Rankin-Selberg zeta function

Can someone explain to me where come from the functional equation (1.6) in this article ?
2
votes
1answer
53 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
2
votes
1answer
493 views

expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
2
votes
1answer
86 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(\...
2
votes
1answer
149 views

Definition of the Weil Group: where is this map $G_1/G_1^c \rightarrow \overline{G}/\overline{G}^c$ coming from?

The following comes from a 1971 paper by R. Langlands, "On the Functional Equation of the Artin L-functions" There seem to be some formatting errors, for example $\mathcal S$ should be replaced by $\...
2
votes
0answers
38 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 ...
2
votes
0answers
31 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 ...
2
votes
0answers
83 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(\...
2
votes
0answers
79 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 ...
2
votes
0answers
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 (...
2
votes
0answers
56 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 ...
2
votes
0answers
68 views

Dirichlet L-function has no zeroes in Re(s) > c implies Euler product converges in Re(s) > c?

If a Dirichlet $L$-function has no zeroes in $\Re(s) \gt c$, does its Euler product necessarily converge in $\Re(s) \gt c$? So I know the proof that (conditional) convergence of the Euler product $\...