# On M-functions associated with modular forms

Abstract : Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or $(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $\chi$ or $f$ vary. First, for a quasi-character $\psi\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit for the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi, s)),$ when $f$ is fixed and $\chi$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes \chi,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h_f \psi(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimes\chi, s).$
Keywords :
Type de document :
Pré-publication, Document de travail
2016

https://hal.archives-ouvertes.fr/hal-01475616
Contributeur : Alexey Zykin <>
Soumis le : jeudi 23 février 2017 - 21:22:46
Dernière modification le : lundi 6 mars 2017 - 11:04:38
Document(s) archivé(s) le : mercredi 24 mai 2017 - 14:26:10

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ihara2017_01.pdf
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### Identifiants

• HAL Id : hal-01475616, version 1
• ARXIV : 1702.07610

### Citation

Philippe Lebacque, Alexey Zykin. On M-functions associated with modular forms. 2016. <hal-01475616>

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