# Would countability conjecture and degree conjecture imply unique factorization for the Selberg class?

Assuming countability conjecture as stated in http://www.mat.unimi.it/users/molteni/research/papers-pdf/4-molteni-Linear_independence_in_the_Selberg_class.pdf and degree conjecture for the whole Selberg class $\mathcal{S}$, one can associate to any element $F$ of $\mathcal{S}$ a positive integer $n(F)$ such that:
i) $n(s\mapsto 1)=1$
ii) $n(F)$ is prime if and only if $F$ is a primitive element of $\mathcal{S}$ of positive degree
iii) $n(F)=n(G)$ if and only if $F=G$
iv) $n(F)$ is prime and $d$ is the least $k$ such that $\pi^{(k)}(n(F))$ is composite if and only if $F$ is primitive of degree $d$ ($d$ will be called the arithmetical degree of $n(F)$)
v) $n(F.G)=n(F).n(G)$
Then, using $deg(F.G)=deg(F)+deg(G)$ one can define the arithmetical degree of $n(F)$ where $F$ is any element of $\mathcal{S}$ as the degree of $F$. Doing this, the Selberg class would be, always assuming the aforementioned conjectures, in a one-to-one correspondence with the set of the positive integers. But the fundamental theorem of arithmetic says that every positive integer greater than $1$ is the product in a unique fashion of prime numbers up to the order. So would the considered "isomorphism" between $\mathcal{S}$ and $\mathbb{Z}_{>0}$ show that the countability conjecture together with the degree conjecture imply that every element of the Selberg class factors in a unique fashion in a product of primitive elements up to the order? Thanks in advance.