Dirichlet \(L\)-Function of Principal Character
Theorem
Let \(\chi_0\) be the principal character modulo \(q\). Then
\[ L(s, \chi_0) = \zeta(s) \prod_{p \mid q} \left(1 - \frac{1}{p^s}\right).\]
Proof
From the Euler product formula for Dirichlet \(L\)-functions when using the principal character our product reduces to primes for which \(\gcd(p, q) = 1\). This is equivalent to when \(p \not\mid q\), and therefore
\[ L(s, \chi_0) = \prod_{p} (1 - \chi_0(p)p^{-s})^{-1} = \prod_{p \not\mid q} (1 - p^{-s})^{-1}\]
given when \(p \mid q\) the term reduces to \(1\) with \(\chi_0(p) = 0\).
Therefore
\[\begin{align*}
\zeta(s) &= \prod_{p} (1 - p^{-s})^{-1} \\
&= \prod_{p \mid q} (1 - p^{-s})^{-1} \prod_{p \not\mid q} (1 - p^{-s})^{-1} \\
&= \prod_{p \mid q} (1 - p^{-s})^{-1} L(s, \chi_0)
\end{align*}\]
and hence
\[ L(s, \chi_0) = \zeta(s) \prod_{p \mid q} \left(1 - p^{-s}\right).\]