# $\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest.

Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where $\phi$ is the euler $\phi$-function.

Then why is $\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{(1-\frac{1}{p^{f(p)s}})^{g(p)}}$ a Dirichlet series with positive integral coefficients?

(The above formula is derived from $L$ functions, $\zeta_m(s)=\prod\limits_\chi L(s,\chi).$)