Dirichlet L-Function of Principal Character

Theorem

Let χ0 be the principal character modulo q. Then

L(s,χ0)=ζ(s)∏p∣q(1−1ps).
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∤q, and therefore

L(s,χ0)=∏p(1−χ0(p)p−s)−1=∏p∤q(1−p−s)−1

given when p∣q the term reduces to 1 with χ0(p)=0.

Therefore

ζ(s)=∏p(1−p−s)−1=∏p∣q(1−p−s)−1∏p∤q(1−p−s)−1=∏p∣q(1−p−s)−1L(s,χ0)

and hence

L(s,χ0)=ζ(s)∏p∣q(1−p−s).