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:


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

where $\backslash$ denotes divisibility.

Proof
By definition:
 * $\chi_0 \left({a}\right) = \begin{cases} 1 & : \gcd \left({a, q}\right) = 1 \\ 0 & : \text{otherwise} \end{cases}$

Therefore:

Hence the result.