Upper and lower bounds at s=1 for certain Dirichlet series with Euler product

摘要

Estimates of the form L-(j) (s, A) much less than(epsilon, j, DA) R-A(epsilon) in the range s - 1 much less than 1/ logR(A) for general L-functions, where R-A is a parameter related to the functional equation of L(s, A), can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the L-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every L(s,pi), where pi is an automorphic cusp form on GL(d, A(K)). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form. [References: 21]