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 $L(f\otimes\chi, s).$
Categories: math.NT