Does $F\otimes G\in\mathcal{M}$?

Question

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ for all prime $p$ where $a_{n}(H)$ is the $n$-th Dirichlet coefficient of $H$, that is $H(s)=\sum_{n>0}\frac{a_{n}(H)}{n^s}$ whenever $\Re(s)>1$.
Does $F\otimes G\in\mathcal{M}$ or are some extra conditions needed? Thanks in advance.