Weil’s converse theorem for Maass forms and cancellation of zeros
Volume 196 / 2020
Acta Arithmetica 196 (2020), 387-422 MSC: 11F66, 11M41, 11F12. DOI: 10.4064/aa190811-3-2 Published online: 11 July 2020
We prove two principal results. Firstly, we characterise Maass forms in terms of functional equations for Dirichlet series twisted by primitive characters. The key point is that the twists are allowed to be meromorphic. This weakened analytic assumption applies in the context of our second theorem, which shows that the quotient of the symmetric square $L$-function of a Maass newform and the Riemann zeta function has infinitely many poles.