Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of L-functions to be any subclass of $\mathcal{L}$, containing the constant map equal to $1$ and the Riemann zeta function, and that is closed under both the usual product $\times$ and the Rankin-Selberg convolution $\otimes$. Say an element $F$ in a Galois class of L-functions $\mathcal{G}$ is RS-primitive in $\mathcal{G}$ if for all pair $(G,G')$ of elements of $\mathcal{G}$, one has $F\neq G\otimes G'$.
Is there a way to order a given Galois class of L-functions $\mathcal{G}$ and an analogue of the PNT giving the proportion of $RS$-primitive L-functions in $\mathcal{G}$ not exceeding the $n$-th one? If so, can this be used to prove that a positive proportion of L-functions belong to the maximal Galois class of L-functions denoted by $\mathcal{M}$?