A Note on Burgess Bound

Let f be a SL(2, Z) Hecke cusp form, and let chi be a primitive Dirichlet character modulo M, which we assume to be prime. We prove the Burgess-type bound for the twisted L-function: L (1/2, f circle times chi) <<(f,epsilon) M1/2-1/8+epsilon. The method also yields the original bound of Burgess for Dirichlet L-functions: L (1/2, chi) <<(epsilon) M1/4-1/16+epsilon.