Dirichlet L-Function from Trivial Character

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

Let $\map L {s, \chi_0}$ be the Dirichlet $L$-function associated to $\chi_0$.

Let $\map \zeta s$ be the Riemann zeta function.

Then:
 * $\ds \map L {s, \chi_0} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$

where $\divides$ denotes divisibility.

Proof
By definition:
 * $\map {\chi_0} a = \begin{cases} 1 & : \gcd \set {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$

Therefore:

Hence the result.