L-Function does not Vanish at One

Theorem
Let $\psi$ be a non-trivial Dirichlet charater modulo $q$.

Let $L \left({s, \chi}\right)$ be the Dirichlet $L$-function associated to $\chi$.

Then $L \left({1, \chi}\right) \ne 0$.

Proof
Let $G^*$ be the group of characters modulo $q$.

Let $\displaystyle \zeta_q \left({s}\right) = \prod_{\chi \mathop \in G^*} L \left({s, \chi}\right)R$.

Among the factors of $\zeta_q$ is the the $L$-function associated to the trivial character, which by Analytic Continuation of Dirichlet L-Function we know to have a simple pole at $s = 1$.

Suppose that $L \left({1, \psi}\right) = 0$.

Then the zero of this factor kills the pole of the principal $L$-function.

So by Analytic Continuation of Dirichlet L-Function $\zeta_q$ is analytic on $\Re \left({s}\right) > 0$.

For $\Re \left({s}\right) > 1$ we have

For any prime $p$, let $f_p$ be the order of $p$ mod $q$.

Then:


 * $\chi \left({p}\right)^{f_p} = \chi \left({p^{f_p}}\right) = \chi \left({1}\right) = 1$

So $\chi \left({p}\right)$ is an $f_p$th root of unity.

Moreover by the Orthogonality Relations for Characters each distinct such root occurs $\phi \left({q}\right) / f_p$ times among the numbers $\chi \left({p}\right)$ where $\chi \in G^*$.

Also, letting $\xi$ be a primitive $f_p$th root of unity we find that for any $u \in \C$:


 * $\displaystyle \prod_{i \mathop = 0}^{f_p} \left({1 - \xi^i u}\right) = 1 - u^\xi$

Putting these facts together:


 * $\displaystyle \prod_{\chi \mathop \in G^*} \frac 1 {1 - \chi \left({p}\right) p^{-s} } = \left({\frac 1 {1 - p^{-f_p s} } }\right)^{\phi \left({q}\right) / f_p}$

Therefore:

Also, if $\chi_0$ is the trivial character modulo $q$, by Euler Product we have:


 * $\displaystyle L \left({\phi \left({q}\right) s, \chi_0}\right) = \prod_{p \mathop \nmid q} \left({1 + p^{-\phi \left({q}\right) s} + p^{-2 \phi \left({q}\right) s} + \cdots}\right)$

from which we see that for $s \in \R_{\ge 0}$:
 * $\zeta_q \left({s}\right) \ge L \left({\phi \left({q}\right) s, \chi_0}\right)$

However, by Analytic Continuation of Dirichlet L-Function, $L \left({\phi \left({q}\right) s, \chi_0}\right)$ diverges for $s = \phi \left({q}\right)^{-1}$, and therefore so does $\zeta_q \left({s}\right)$.

But we showed above that $\zeta_q \left({s}\right)$ converges for $\Re \left({s}\right) > 0$, a contradiction.