# Questions tagged [l-functions]

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

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$L-$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 $$L_p(E,s) = \begin{cases}(1-a_pp^... 0answers 15 views ### Dirac Measure and Iwasawa Algebras I am reading through some lecture notes on p-adic L-functions, and one of the exercises asks for a \in \mathbb{Z}_p that we define the Dirac measure \delta_a by \int_{\mathbb{Z}_p} \phi \cdot \... 0answers 24 views ### Limits with zeroes of L-functions As usual, let L(s,\chi) = \sum_{n\geq 1} \chi(n)n^{-s} be the L-function of a Dirichlet character mod q and assume GRH. Since no nonzero meromorphic function has infinitely many zeroes in a ... 0answers 149 views ### Proof a 'well-known' result of Shimura on periods of modular forms It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic.... 1answer 41 views ### Special values of Hecke L-function on imaginary quadratic fields Let K be an imaginary quadratic number field and \mathcal{O}_K its ring of integers. Let \chi be an algebraic Hecke character on K with conductor \mathfrak{f} and infinity type (a,b), i.e. ... 0answers 81 views ### Question on paper of Mazur, Tate, Teitelbaum and p-adic L functions of modular forms I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following. Let f be a cuspidal eigenform of weight k ... 0answers 49 views ### Euler Factors in Permutation Representation of Galois Group Let k be a number field and K / \mathbb Q a Galois extension containing k, with Galois group G=\operatorname{Gal}(K/\mathbb Q) and let G_k:=\operatorname{Gal}(K/k). Let \chi denote the ... 1answer 22 views ### Order of vanishing of L function In the introduction of this paper, the authors say that: Let M \in \mathbb{Z}_{>0}. If f is a normalised newform for \Gamma_0(M) then we define$$\Lambda(s,f)=(2\pi)^{-s}\Gamma(s) M^{s/2}L(s,...
I have seen examples given of $L$-functions, such as Dirichlet $L$-functions and the Riemann Zeta Function, but I have not seen a definition of the most general form of an $L$-function. Basically what ...