ON M-FUNCTIONS ASSOCIATED WITH MODULAR FORMS

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 L(f circle times chi, s) denote either log L(f circle times chi, s) or (L'/L)(f circle times chi, s). In this article we study the distribution of the values of ,Z when either chi or f vary. First, for a quasi-character psi: C -> C-x we find the limit for the average Avg(x) psi(L(f circle times 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 L(f circle times chi, s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avg(f)(h) psi(L(f , s)), when f runs through the set of primitive cusp forms of given weight k and level N -> infinity. Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for L(f circle times chi, s).