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## On M-functions associated with modular forms

HAL:archives-ouvertes. HAL. Le Centre pour la Communication Scientifique Directe, 2017
Lebacque P., Zykin A. I.
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 equidistribu tion 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 ) .