## A converse theorem for degree 2 elements of the Selberg class with restricted gamma factor

### Volume 205 / 2022

Acta Arithmetica 205 (2022), 41-52
MSC: Primary 11M36; Secondary 11M41.
DOI: 10.4064/aa211001-27-6
Published online: 8 September 2022

#### Abstract

We prove a converse theorem for a family of L-functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L-functions coincide with either those coming from a newform or a product of L-functions arising from Dirichlet characters. We require some analytic data on the Euler factors, but do not require anything on the shape. We also suppose that the twisted L-functions satisfy expected functional equations. We incorporate the ideas from a 2014 paper by Booker and Krishnamurthy so that the non-trivial twists are allowed to have arbitrary poles.