TWISTS OF AUTOMORPHIC L-FUNCTIONS AT THE CENTRAL POINT

Abstract. We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character χ modulo D satisfying some technical condi-tions, we prove that the twisted L-functions L(f.χ, s) do not vanish at s = 1 2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.χ, s), and from that derive an upper bound for the average analytic rank of the studied L-functions. 1