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

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Do UF+PNT+SMO+GRH imply SOC?

The title may sound esoteric, but let's make it explicit. Suppose that the conjunction of unique factorization (UF, still open), prime number theorem (PNT, proved by Yoshikatsu Yashiro), Strong ...
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Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
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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 ...
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Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil $L$ function. Is the rank of the abelian group $E(K)$ of points of $E$ the ...
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Does $F\otimes G\in\mathcal{M}$?

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ ...
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Analytic formulas for special values of $L$-functions

In The Princeton Companion to Mathematics, IV.1, “Algebraic Numbers”, the conditionally convergent series \begin{equation} ...
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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, ...
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Multiplicity one theorem for GL(n) and SL(n) [closed]

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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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 ...
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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 ...
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is the degree of an L-function a semiring homomorphism?

Consider the biggest subset of the intersection of the Selberg class and the set of automorphic L-functions closed under product and tensor product (that corresponds on the automorphic side to ...
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42 views

dimension of a scheme and degree of an L-function

Disclaimer: I first asked this question on Mathoverflow but I was told it was off-topic for that site, so I ask it here. I try to understand correctly the notion of scheme, as Serre in the second ...
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38 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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28 views

Effective bounds on L(1,Chi) for Chi a Dirichlet Character

I have $\chi$, a Dirichlet Character $\bmod n$, and I have established that $L(1,\chi) \geq C / \log \log n$ for some constant C under the generalized Riemann Assumption. I'll call this proposition ...
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47 views

Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
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a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
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On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
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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 ...
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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 ...
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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 ...
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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 ...
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$L$-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 ...
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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 ...
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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 ...
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Why are L-functions called such?

L-functions are discussed here: http://www.lmfdb.org/intro/tutorial... So why are they called L-functions...? Is there a reason for the L?
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Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip
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106 views

Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
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On the special value of Hecke L function.

For a nontrivial Hecke character $\chi:A_Q^{\times}/Q^{\times}\to S^1$, we know $L_Q(s,\chi)$ is nonzero. Is this true for number field $F$? I know is is holomorphic at $s=1$ by Artin conjecture, but ...
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119 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 ...
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107 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. ...
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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( ...
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$L$-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 $F$. ...
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Base-Change for $GL_1$

I try to understand the base-change in the theory of automorphic forms for the simplest case: $GL_1$. Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of ...
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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 ...
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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) = ...
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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 ...
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Riemann-Weil formula question

given a sum over the imaginary part of the zeros $ \sum_{t} h(t) $ is this version of the Riemann-Weyl formula correct? $ \sum _{t} h(t) -2h(i/2)= -2 \sum_{n=1}^{\infty}\Lambda (n)g(\log ...
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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 ...
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531 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.