# Moscow Mathematical Journal

Volume 18, Issue 3, July–September 2018 pp. 437–472.

On *M*-Functions Associated with Modular Forms

**Authors**:
Philippe Lebacque (1) and Alexey Zykin (2)

**Author institution:**(1) Laboratoire de Mathématiques de Besançon, UFR Sciences et techniques 16, route de Gray 25 030 Besançon, France

(2) Laboratoire GAATI, Université de la Polynésie française,
BP 6570 – 98702 Faa'a, Tahiti, Polynésie française

National Research University Higher School of Economics

AG Laboratory NRU HSE

Institute for Information Transmission Problems of the Russian Academy of Sciences

**Summary: **

Let *f* be a primitive cusp form of weight *k* and level *N*,
let χ be a Dirichlet character of conductor coprime with *N* , and let
𝔏(*f*⊗χ, *s*) denote either log *L*(*f*⊗χ, *s*) or (*L*′/*L*)(*f*⊗χ, *s*). In this article
we study the distribution of the values of *L* when either χ or *f* vary.
First, for a quasi-character ψ: ℂ → ℂ^{×} we find the limit for the average
Avg_{χ}ψ(*L*(*f*⊗χ, *s*)), when *f* is fixed and χ varies through the set of
characters with prime conductor that tends to infinity. Second, we prove
an equidistribution result for the values of 𝔏(*f*⊗χ, *s*) by establishing
analytic properties of the above limit function. Third, we study the
limit of the harmonic average Avg* ^{h}_{f}* ψ(

*L*(

*f*,

*s*)), when

*f*runs through the set of primitive cusp forms of given weight

*k*and level

*N*→ ∞. Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for

*L*(

*f*⊗χ,

*s*). 2010 Math. Subj. Class. Primary: 11F11; Secondary: 11M41.

**Keywords:**

*L*-function, cuspidal newforms, value-distribution, density function.

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