$L(s,\chi)=1$ for non trivial Dirichlet character modulo a polynomial of degree $1$

Question

Let $m(x)\in \Bbb F_q[X]$ be a monic polynomial of degree $1$.

Show that for every non-trivial Dirichlet character modulo $m$ we have: $L(s,\chi)=1$.

I have seen a theorem that states that if $\chi$ is non trivial, then $L(s,\chi)$ is a polynomial in $q^{-s}$ of degree at most $deg(m)-1$. In my case, it means the degree is $0$, so $L(s,\chi) \in \Bbb F_q$. I still can't understand why it equals $1$.

Answer 1

$$L(T,\chi) = \sum_{f \in F_q[X]_{monic}} \chi(f) T^{\deg(f)} = \\ T^{\deg(1)} + \sum_{a \in F_q[X]/(m)} \chi(a) \sum_{f \in F_q[X]_{monic}} T^{\deg(f m +a)}$$ Can you finish from there ?