Twists, Euler products and a converse theorem for $L$-functions of degree 2 in the Selberg class

We prove a general result relating the shape of the Euler product of an $L$-function to the analytic properties of certain linear twists of the $L$-function itself. Then, by a sharp form of the transformation formula for linear twists, we check the required analytic properties in the case of $L$-functions of degree 2 and conductor 1 in the Selberg class. Finally we prove a converse theorem, showing that $\zeta(s)^2$ is the only member of the Selberg class satisfying the above conditions and, moreover, having a pole at $s=1$.