Which properties of L-functions can be proven assuming they are objects of a symmetric bimonoidal category?

Question

The title says it all : assuming all L-functions are objects of a symmetric bimonoidal category $ (\mathcal{C},\oplus,\otimes,s\mapsto 1,\zeta) $ , where $ \oplus $ stands for the usual product, what can we prove about L-functions from general theorems about symmetric bimonoidal categories that turns out to have analytic or number theoretic interest ? I know some people here might get angry to see me ask this question but still, it may be insightful.

Answer 1

Let $\mathcal{D} = \{ \sum_{n=1}^\infty a_n n^{-s} = \prod_{j=1}^l L(s,\chi_j)^{e_j}, e_j \in \mathbb{Z}\}$ the set of products and quotients of Dirichlet L-functions and $$T : \mathcal{D} \to \ldots, \qquad T(\sum_{n=1}^\infty a_n n^{-s}) = \sum_p a_p p^{-s}$$ By multiplicate-ness $\scriptstyle\sum_{n=1}^\infty a_n n^{-s} =\prod_p (1+\sum_{k \ge 1}a_{p^k} p^{-sk})= \prod_p (\prod_{j=1}^l (1-\chi_j(p) p^{-s})^{-e_j})$ $\scriptstyle= \prod_p (\prod_{j=1}^l (1+\chi_j(p) p^{-s}+\mathcal{O}(p^{-2s}))^{e_j}) =\prod_p (\prod_{j=1}^l (1+e_j \chi_j(p) p^{-s}+\mathcal{O}(p^{-2s})))$ $\scriptstyle=\prod_p (1+\sum_{j=1}^l e_j \chi_j(p) p^{-s}+\mathcal{O}(p^{-2s}))$ thus $a_p = \sum_{j=1}^l e_j \chi_j(p)$ and $$T(\prod_{j=1}^l L(s,\chi_j)^{e_j}) = \sum_p p^{-s} \sum_{j=1}^l e_j \chi_j(p)$$

Thus $T\mathcal{D}$ is an abelian group under addition, and exploiting the fact products of Dirichlet characters are Dirichlet characters, there is a natural ring structure on $T\mathcal{D}$ :

$$(\sum_p p^{-s} \chi_1(p))\times (\sum_p p^{-s} \chi_2(p)) =\sum_p p^{-s} \chi_1(p)\chi_2(p)$$

Then we quotient $T \mathcal{D}$ by the equivalence relation "equal on all but finitely many primes". We then obtain that $$T \mathcal{D} = \{ \sum_p p^{-s} \sum_{j=1}^l e_j \chi_j(p), \quad e_j \in \mathbb{Z},\quad \chi_j \text{ primitive Dirichlet charaters}\}$$ Almost by definition, the set of primitive Dirichlet characters is a group $G\cong \lim_{n \to \infty} (\mathbb{Z}/n!\mathbb{Z})^\times$ thus $T \mathcal{D} = \mathbb{Z}[G]$ as a ring (and it is a subring of $\mathbb{Z}[\hat{\mathbb{Z}}^\times]$ whatever it means in term of L-functions)

The question is if all this works when assuming some Langlands conjectures and replacing the Dirichlet L-functions by the automorphic normalized L-functions $L(s,\pi)$, I'd say yes, since $T (L(s,\pi_1)) \times T(L(s,\pi_2)) = T(L(s,\pi_1 \otimes \pi_2))$.

What would be the ring structure of $T \mathcal{L}$ ? How would you define $T^{-1}$ ?