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      Questions tagged [l-functions]

      Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.

      4
      votes
      2answers
      117 views

      Real non trivial zeros of Dirichlet L-functions

      When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
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      votes
      0answers
      114 views

      Second derivative at 1 of L function of elliptic curve

      Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that $$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$ where $\gamma$ is Euler's ...
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      votes
      1answer
      110 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. 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 the ...
      7
      votes
      0answers
      196 views

      A mysterious number related to Hasse-Weil L-function of elliptic curve

      Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
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      votes
      2answers
      76 views

      Why we use Caputo fractional derivative in application?

      I'm working on some papers which use Caputo fractional evolution equation as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{...
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      votes
      0answers
      121 views

      How many exceptional conductors are there?

      We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
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      votes
      1answer
      236 views

      Functional equation for general number fields

      When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
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      votes
      0answers
      86 views

      Function equation over general number fields

      Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
      4
      votes
      1answer
      195 views

      Modular forms and Period Polynomials

      1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is ...
      3
      votes
      0answers
      169 views

      holomorphic continuation of motivic $L$-functions

      The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
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      vote
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      42 views

      Mean value estimates for general number fields

      Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...
      7
      votes
      1answer
      308 views

      Relation between Fourier coefficients and Satake parameters

      Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part: $$L(s) =...
      4
      votes
      1answer
      101 views

      Corollary for Casselman-Shalika formula

      Assume $\pi$ is an unramified representation of $GL_n(F)$, where $F$ is a p-adic field. And $\phi$ is an unramified vector for $\pi$. Assume $W_{\phi}$ is a Whittaker function associated to $\phi$. ...
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      votes
      1answer
      477 views

      Euler factors of L-function at bad primes

      This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including ...
      8
      votes
      1answer
      220 views

      Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties

      $\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} \newcommand{\p}{\mathfrak{p}} $ Let $A$ be an abelian ...

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