UPPER AND LOWER BOUNDS AT s = 1 FOR CERTAIN DIRICHLET SERIES WITH EULER PRODUCT

Estimates of the form L(s,A) ,j,DA R A in the range |s− 1| 1/ logRA for general L-functions, where RA 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, π), where π is an automorphic cusp form on GL(d,AK). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form. Mathematics Subject Classification (2000): 11M41 1 Definitions and results We consider the class of functions satisfying the following axioms: (A1) (Euler product) Let A = {Ap}p, p prime, be a sequence of complex square matrices of order d, with monic characteristic polynomial Pp(x) = PA p (x) ∈ C[x] and roots αj(p) = αA j (p). We define the general L-function L(s,A) as