Zeros of $L$-functions outside the critical strip

Abstract

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if $f\in S_k\left({\Gamma }_1\left(N\right)\right)$ is a classical holomorphic modular form whose $L$-function does not vanish for $\Re \left(s\right)>\left(k+1\right)∕2$, then $f$ is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree-$1$ $L$-functions.