  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
$s=1$.
