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 is a classical holomorphic modular form whose L-function does not vanish for Re(s) > (k+1)/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.

Includes a footnote, not appearing in the published paper, summarizing a (minor) correction to appear in the same journal

References in corpus (1)

• Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

Cited by in corpus (8)

• L-functions as distributions
• Zeros of combinations of Euler products for $\sigma >1$
• On the density of zeros of linear combinations of Euler products for $\sigma >1$
• Joint universality on the half plane of absolute convergence
• On the zeros of generalized Hurwitz zeta functions
• Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence
• Zeros of $L(s)+L(2s)+\cdots +L(Ns)$ in the region of absolute convergence
• Combinations of $L$-functions and Their Non-coincident Zeros for $\sigma >1$