Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
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What's the idea of Dirichlet’s Theorem on Arithmetic Progressions proof?
Dirichlet’s Theorem on Arithmetic Progressions says that if $a, m$ are natural numbers such that $gcd (a,m) = 1$, then there are infinitely many prime numbers in the arithmetic progression $a + km, k \...
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1
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On the Iwasawa Algebra
I am reading Joaquin Rodrigues Jacinto's and Chris Williams' notes on $p$-adic $L$-functions http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Number-Theory---Full-Lecture-Notes-2017-...
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Definition $L$ function of elliptic curve
Let $E$ be an elliptic curve.
My book reads, we denote '$L$ series of $E$ regrading $l$ adit representation of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$' as
$L(E/\Bbb{Q}):= \prod_{p}\frac{1}{1-a_pp^{-s}+...
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Normalisation of $L$-function for classical modular forms and automorphic representations
I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a ...
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First few coefficients of $\zeta_p$ as an element of an Iwasawa algebra
One way of introducing the $p$-adic Riemmann zeta function is to first define a $p$-adic pseudomeasure $\zeta_p$ via interpolation. Namely, $\zeta_p$ is uniquely defined by the property that
$$\int_{\...
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2
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Selberg Class- Ramanujan Conjecture
The wikipedia for Selberg class L-functions (https://en.wikipedia.org/wiki/Selberg_class)
states 4 conditions:
Analyticity,
Ramanujan conjecture,
Functional equation,
Euler product.
I would like to ...
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Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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Do $L$-functions of varieties over function fields have analytic continuation / functional equations?
Suppose that $K$ is a function field (i.e: a finite extension of $\mathbf{F}_q(t)$) and $X$ is a smooth projective variety over $K$. I have a few questions about the $L$-function of $X/K$:
How do you ...
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Root Number is root of unity
In his thesis Tate deduces a functional equation for Hecke L-Functions associated to a unitary idele class character $\chi$ of the form
$$ L(s,\chi)=\epsilon(s,\chi)L(1-s,\overline{\chi}).$$
Here the ...
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Interpretation of order of vanishing of Modular L-function
The BSD conjecture gives an interpretation to the order of vanishing of the central value the L-function of an elliptic curve (it is supposed to be the rank of that elliptic curve), while the ...
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What is the relationship between these two versions of BSD?
The BSD conjecture is usually formulated like this.
If $E/\mathbf{Q}$ is an elliptic curve, then
$$ \text{rank }E/\mathbf{Q} = \text{ord}_{s=1} L(E,s), $$
where $L(E,s)$ is the Hasse-Weil $L$-...
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0
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Proving that order of pole of L-function is greater than or equal to 0 by Artin's Reciprocity Law
I have been studying the proof of Chebotarev's Density Theorem from the book "Problems in Algebraic number Theory" by Ram Murty and Esmonde and encountered a problem.
Let $n(H, \psi)$ be the ...
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One identity about Eisenstein series and L functions of modular forms
Currently, I'm reading Zagier's notes 'Modular forms whose Fourier coefficients involve zeta-function of quadratic fields'. There is an equation (72) in page 42:
$$\sum_{n=1}^{\infty} \frac{\sigma_{r}(...
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1
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A question on the Riemann zeta function
Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ...
user1067130
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1
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Approximate functional equation of Selberg Class L-functions
I was reading the paper "Integral moments of $L$-functions" by Conrey et al. They have used the following sharp cutoff version of the approximate functional equation for "Selberg class&...
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1
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Whittaker at Archimedean Test vector
Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression ...
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Why are L-factors polynomials in $q^{-s}$?
I am interested on integrals of "Tate style", say of the form
$$\int_{F^\times} \Phi(a) \kappa(a) |a|^s d^\times a$$
Here, $F$ is a local field, $|\cdot|$ the associated absolute value, $\...
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Davenport, the usage of mean value theorem
In Davenport, Multiplicative Number Theory, page 6, the author asserts that:
"Let $\omega$ be a complex character. Suppose $L_{\omega}(1)=0$, then it would imply that $L_{\omega}(s)=L_{\omega}(s)-...
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Why are Rankin-Selberg convolutions different when $n=m$?
Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
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What is the importance of RH for $L$-functions of modular forms?
The Reimann Hypothesis (RH) for $L$-functions of modular forms states that all the non-trivial zeroes of an $L$-function of a modular form must lie on the critical line. My question is: why is this ...
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*Why* the converse theorem (conjecture) should be true?
Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...
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Logarithmic average of the Legendre symbol
My question is simple: can we show that the sum
$$ \sum_{k=1}^{p-1} \frac{\left( \frac{k}{p} \right)}{k} $$
is positive for all primes $ p $ where $ (k/p) $ denotes the Legendre symbol modulo $ p $, i....
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Relations between different zeta functions for a simple algebra
I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
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0
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Triangle inequality of Dirichlet cofficients of Artin L-function.
As for automorphic L-function we know that
$$
\left|\lambda_{\pi}(n)\right|^{2} \leq \lambda_{\pi \times \widetilde\pi}(n)
$$
is there exist a inequality similarly for Artin L-functions?
For above ...
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Why is $s=1$ in Dirichlet's Class Number Fomula when $L$-functions are only defined for $\Re(s)>1$?
Given a quadratic number field $K$, let $w$ be the roots of unity, $d_K$ the discriminant, $h$ the class number and let $\varepsilon$ be the fundamental unit of $O_K$. We have that Dirichlet's Class ...
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Proposition 10.4 in Neukirch's Algebraic Number Theory
I am stuck on a small detail in the proof of Proposition 10.4(iii) from Chapter VII of Neukirch's Algebraic Number Theory.
For a Galois extension of number fields $L|K$ and a representation $(\rho, V)$...
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Looking for table of special values of the Dirichlet $L$-function
For double checking calculations I made I'd like to find a table of values of $L(-1,\chi_D)$ for small positive fundamental discriminats $D$. It there a table somewhere in the internet? Where?
With $\...
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Artin $L$-function is well defined
Here is what I got so far. Let $\rho: Gal(L/K)\rightarrow GL(V)$ be an Artin representation. Let $\frak{p}$ be a prime of $K$, $\frak{P}$ a prime of $L$ lying above $\frak{p}$ and denote by $D_\frak{...
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Residue of Rankin Selberg L-function
Let $f$ be a normalized holomorphic cusp form with weight $k$, level $N$. The Fourier expansion of $f$ can be written as
\begin{align*}
f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{(k-1)/2} e^{2\pi inz}
\...
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Meromorphic continuation of L-functions
I am following these notes and on page 2 the claim is that if we have an $L$-function $$L(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ with $a_n=O(n^r)$ and if $L$ has a meromorphic continuation and ...
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Approximate functional equation of the L function
I was reading a paper on Selberg's central limit theorem for the classical automorphic $L$ functions attached to primitive holomorphic cusp form $f$. I can not understand the following equation.
\...
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Factorization of $L$-functions for CM Elliptic Curves
I saw recently that the $L$-functions of elliptic curves with CM can be factored as a product of simpler $L$-functions. In this question, I'd like to ask why that factorization is significant and what ...
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Searching for a concept/clue: efficient algorithm to extract columns in a large table/matrix with minimal collisions: a use case of class field theory
Although the algorithmic problem is very generic in nature and there are many (possibly more down to earth) examples of its application, I do not want to suppress the real context of my problem.
I ...
user736865
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votes
1
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Controlling High Moments with Short Low Moments
Recently I've been working on a project for which I've needed to reference Iwaniec's paper Fourier coefficients of cusp forms and the Riemann zeta function, where the short fourth moment estimate
$$
\...
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1
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Functional equation for Rankin-Selberg L functions in the imprimitive case
If $f$ and $g$ are primitive modular forms of characters $\chi$ and $\psi$, such that $\chi, \psi$ and $\chi * \psi$ are all primitive, then we have an explicit functional equation. This is proven in ...
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2
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Is there a Kubota-Leopoldt $p$-adic zeta function implementation in SageMath?
Exactly as in the title. I am learning the $\mathbb{Z}_p^\times$ measure-theoretic construction of $p$-adic $L$-functions and was wondering if there was an `easy' way produce some example computations ...
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$L$-function of elliptic curves expansion into Dirichlet series
Let $E/\mathbb{Q}$ be an elliptic curve. The $L$-function of $E$ is defined to be the Euler product
$$
L_E(s) = \prod_{\text{ bad }p} (1 - a_p p^{-s})^{-1} \prod_{\text{ good }p} (1 - a_p p^{-s} + p^{...
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Dirichlet L-series and Hecke L-series
I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?
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L-function of Hilbert modular forms of non-parallel weight
I found that in many literature, they only define the L-function of Hilbert modular forms of parallel weight. Can we define the following L-function:
\begin{equation}
L(f;s_1,\cdots,s_n)=\int_0^{i\...
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1
answer
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Counting function and its Artin L-function
I am working with the following function: given a polynomial, $P(x)$, with non-negative integer coefficients and passing through origin, we can define
$$f(d)=\{1\leq a\leq d \ | \ P(a)\equiv 0 \mod(d)\...
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L-Function of Elliptic Curve and Modular Form
I have a problem to see how the galois-theoretic definition of the L-function of an elliptic curve gives the right answer and also the connection between the L-function of a weight 2 Hecke eigenform ...
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Why is it necessary to find extreme values of L-functions?
Apparently I have been exploring Riemann Zeta Function and have lately come across $L$-functions. After going through few papers, I realised that many mathematicians are giving their full time knowing ...
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References for this function: $\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}?$
Consider $$\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}$$
where $\chi$ is a Dirichlet character.
Is anything known about it?
I looked around and didn't see any references involving this ...
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Root number of the $L$-function of $y^2 = x^3 - n^2x$ and $n \pmod 8$.
Root number definition. Let $E_n$ be the elliptic curve $y^2 = x^3 - n^2 x$ where $n$ is a positive squarefree integer. It is known that the $L$-function of $E_n$, denoted $L(E_n,s)$, can be extended ...
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An $L-$function and a $J-$function. Related?
Consider a Dirichlet series for a non real character of modulus $q$
$$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$
and $s\in\Bbb C$ with real part greater than one.
Consider a $J$-series $$ J(s,...
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Definition of $L$-function of elliptic curves
$L-$function of elliptic curves is Dirichlet series and defined to be $$
L(E,s) = \sum_{n\ge 1}\frac{a_n}{n^s} = \prod_p L_p(E,s),
$$where the Euler factor at $p$ is
$$
L_p(E,s) = \begin{cases}(1-a_pp^...
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Limits with zeroes of L-functions
As usual, let $L(s,\chi) = \sum_{n\geq 1} \chi(n)n^{-s}$ be the L-function of a Dirichlet character mod $q$ and assume GRH. Since no nonzero meromorphic function has infinitely many zeroes in a ...
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Special values of Hecke $L$-function on imaginary quadratic fields
Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Let $\chi$ be an algebraic Hecke character on $K$ with conductor $\mathfrak{f}$ and infinity type $(a,b)$, i.e.
...
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Question on paper of Mazur, Tate, Teitelbaum and $p$-adic L functions of modular forms
I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following.
Let $f$ be a cuspidal eigenform of weight $k$ and ...
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