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We prove a converse theorem for a family of L functions of degree 2 withgamma factor coming from a holomorphic cuspform. We show these L functionscoincide with either those coming from a newform or a product of L functionsarising from Dirichlet characters. We require some analytic data on the Eulerfactors, but don't require anything on the shape. We also suppose that thetwisted L functions satisfy expected functional equations. We allow thenon-trivial twists to have arbitrary poles.

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