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**Moments of quadratic twists of modular L-functions(****arXiv****)**

**Author : **Xiannan Li

**Abstract : **We prove an asymptotic for the second moment of quadratic twists of a modular L-function. This was previously known conditionally on GRH by the work of Soundararajan and Young.

**2.Notes on the arithmetic of Hecke L-functions (****arXiv****)**

**Author : **A. Raghuram

**Abstract : **This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of the article, on the ratios of successive critical values of the Hecke L-function attached to an algebraic Hecke character, concerns certain variations on a theorem of Guenter Harder, especially drawing attention to a delicate signature that apparently has not been noticed before.

**3.On the vanishing of twisted L-functions of elliptic curves over rational function fields (****arXiv****)**

**Author : **Antoine Comeau-Lapointe, Chantal David, Matilde Lalin, Wanlin Li

**Abstract : **We investigate in this paper the vanishing at s=1 of the twisted L-functions of elliptic curves E defined over the rational function field Fq(t) (where Fq is a finite field of q elements and characteristic ≥5) for twists by Dirichlet characters of prime order ℓ≥3, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li and Donepudi — Li who proved vanishing at s=1/2 for infinitely many Dirichlet L-functions over Fq(t) based on the existence of one, and we can prove that if there is one χ0 such that L(E,χ0,1)=0, then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over Fq(t) behave differently than the general ones.

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