  Given a finite group $G$, we study the monomial algebra $R_G$, generated by
the monomial characters of $G$. In particular, we note that the integral
closure of $R_G$ is contained in the algebra generated by those characters
$\chi$ for which their associated Artin L-function $L(s,\chi)$ is holomorphic
at $s_0\in\mathbb C\setminus\{1\}$. Also, we discuss the supercharacter
theoretic case.
