Special values of symmetric power L-functions and Hecke eigenvalues

Abstract

We compute the moments of L-functions of symmet- ric powers of modular forms at the edge of the critical strip, twisted by the central value of the L-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L-functions. We deduce information on the size of sym- metric power L-functions at the edge of the critical strip in sub- families. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the si- multaneous extremality conditions on the values of L-functions of symmetric powers of modular forms at the edge of the critical strip.