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Abstract

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