A metric approach to zero-free regions for $L$-functions

For integers $m,m^{\mathrm{\prime }}\ge 1$, let $\pi$ and ${\pi }^{\mathrm{\prime }}$ be cuspidal automorphic representations of $\mathrm{G}\mathrm{L}(m)$ and $\mathrm{G}\mathrm{L}(m^{\mathrm{\prime }})$, respectively. We present a new proof of zero-free regions for $L(s,\pi )$ and for $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ under the assumption that $\pi ,{\pi }^{\mathrm{\prime }}$ or $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ is self-dual. Our approach builds on ideas of “pretentious” multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi.