On the semigroup ring of holomorphic Artin L-functions

Let $K\mathrm{/}\mathbb{Q}$ be a finite Galois extension and let ${\chi }_1,\dots ,{\chi }_r$ be the irreducible characters of the Galois group $G:=Gal(K\mathrm{/}\mathbb{Q})$. Let $f_1:=L(s,{\chi }_1),\dots ,f_r:=L(s,{\chi }_r)$ be their associated Artin L-functions. For $s_0\in \mathbb{C}\setminus \{1\}$, we denote $Hol(s_0)$ the semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\mathbb{F}$ be a field with $\mathbb{C}\subseteq \mathbb{F}\subseteq {\mathcal{M}}_{<1}:=$ the field of meromorphic functions of order $<1$. We note that the semigroup ring $\mathbb{F}[Hol(s_0)]$ is isomorphic to a toric ring $\mathbb F[H(s_0)]\subseteq \mathbb F[x_1,\ldots,x_r]$,where$H(s_0)$isanaffinesubsemigroupof$\mathbb N^r$minimallygeneratedbyatleast$r$elements,andwedescribe$\mathbb F[H(s_0)]$whenthetoricideal$I_{H(s_0)}=(0)$\mathrm{.}Also,wedescribe$\mathbb F[H(s_0)]$and$I_{H(s_0)}$when$f_1,\ldots,f_r$ have only simple zeros and simple poles at $s_0$.