Twists, Euler products and a converse theorem for L-functions of degree 2

We prove a general result relating the shape of the Euler product of an L-function to the analytic properties of the 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 the square of the Riemann zeta function is the only member of the Selberg class with degree 2, conductor 1 and a pole at s=1.