Dirichlet L-Function from Trivial Character

Theorem
Let $\chi_0$ be the trivial Dirichlet character modulo $q$.

Let $\zeta$ be the Riemann zeta function.

Then


 * $\displaystyle L(s,\chi_0) = \zeta(s) \cdot \prod_{p\mid q}\left(1 - p^{-s}\right)$

Proof
By definition, $\chi_0(a) = 1$ if $\gcd(a,q) = 1$, and $\chi_0(a) = 0$ otherwise.

Therefore,

As required.