Is the twist of a primitive L-function itself primitive?

Question

Suppose $F$ is a primitive function of the Selberg class and $G$ a Dirichlet L-function. Is the Rankin Selberg convolution $F\otimes G$ itself primitive? This is true if $F$ is itself a Dirichlet L-function, since it has degree $1$ and so has the resulting L-function. But what do we know for primitive L-functions of degree greater than $1$? Thanks in advance.

Answer 1

This is an important open problem. It is sometimes called Selberg's Twist Conjecture, and essentially no progress has been made on this problem, even for degree $2$ $L$-functions.