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Upper and lower bounds at \(s=1\) for certain Dirichlet series with Euler product. (English) Zbl 1100.11028

Summary: Estimates of the form \(L^{(j)}(s,{\mathcal A})\ll_{\epsilon,j,{\mathcal D}_A}{\mathcal R}^\epsilon_{\mathcal A}\) in the range \(|s-1|\ll 1/\log {\mathcal R}_A\) for general \(L\)-functions, where \({\mathcal R}_A\) is a parameter related to the functional equation of \(L(s,{\mathcal A})\), 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 \(\text{GL}({\mathbf d},\mathbb {A}_K)\). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.

MSC:

11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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