Call $L$-function any element of an L-rig (see [Are there infinitely many L-rigs?](https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs) for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does the primitivity of $F\otimes G$ imply the primitivity of both $F$ and $G$?


Edit: perhaps I should add some motivation. I learnt that an irreducible representation of the direct product of two finite groups $G\_{1}$ and $G\_{2}$ is the tensor product of irreducible representations of the respective groups, so drawing and analogy between irreducible representation of a finite group and primitive L-function, I came to think about this question.



Does Rankin-Selberg convolution preserve primitivity?
Let $L(s,F)$ be the $L$-function of a self-dual $\mathrm{GL}(2)$ holomorphic cuspidal newform without complex multiplication and with trivial nebentypus. Let $G=\mathrm{Sym}^2 F$ be the symmetric square lift of $F$. Then $L(s,F)$ is a primitive $\mathrm{GL}(2)$ $L$-function (due to Hecke) and $L(s,G)$ is a primitive $\mathrm{GL}(3)$ $L$-function (due to Gelbart and Jacquet), but


$$L(s,F\otimes G) = L(s,F)L(s,\mathrm{Sym}^3 F),$$


which is not primitive. Simpler and more complicated examples abound; this seemed like a nice example from the middle ground.


