  In a previous paper we proved that if an $L$-function $F$ from the Selberg
class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly
twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler
product. In this paper we extend this result to $L$-functions of degree 2 with
square-free conductor $q_F$, which are weakly twist-regular at all primes
$p\nmid q_F$
