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

Estimates of the form L(j)(s, script A sign) ≪ε,j,script D sign script A signℛεscript A sign in the range |s - 1| ≪ 1/log ℛscript A sign for general L-functions, where ℛscript A sign, is a parameter related to the functional equation of L(s, script A sign), 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, double-struck A signK). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.