Power sums and Siegel-type zero-free regions for L-functions

Let $\pi$ and ${\pi }^{\mathrm{\prime }}$ be unitary cuspidal automorphic representations of $\mathrm{G}\mathrm{L}(n)$ and $\mathrm{G}\mathrm{L}(n^{\mathrm{\prime }})$ over a number field $F$. Let ${\mathfrak{C}}_{\pi }$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\epsilon >0$ the existence of ineffective constants $c=c_{n,F,\epsilon }>0$ and $c^{\mathrm{\prime }}=c_{n,F,{\pi }^{\mathrm{\prime }},\epsilon }^{\mathrm{\prime }}>0$ such that the standard $L$-function $L(s,\pi )$ satisfies $$\mathrm{\mid }L(\sigma +it,\pi )\mathrm{\mid }\ge c({\mathfrak{C}}_{\pi }(\mathrm{\mid }t\mathrm{\mid }+3){)}^{-\epsilon },\sigma \ge 1-c({\mathfrak{C}}_{\pi }(\mathrm{\mid }t\mathrm{\mid }+3){)}^{-\epsilon }$$ and the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ satisfies $$\mathrm{\mid }L(\sigma +it,\pi \times {\pi }^{\mathrm{\prime }})\mathrm{\mid }\ge c^{\mathrm{\prime }}({\mathfrak{C}}_{\pi }(\mathrm{\mid }t\mathrm{\mid }+3){)}^{-\epsilon },\sigma \ge 1-c^{\mathrm{\prime }}({\mathfrak{C}}_{\pi }(\mathrm{\mid }t\mathrm{\mid }+3){)}^{-\epsilon }\mathrm{.}$$ Applications include improvements to the prime number theorems for these $L$-functions and new generalizations of the Brauer-Siegel theorem.