Dirichlet $L$-function of primitive character in function field setting Let $q=p^k$ be a prime power, and let $Q \in \mathbb{F}_q[t]$ be a polynomial. A Dirichlet character $\varphi$ of modulus $Q$ is a group homomorphism
$$
\varphi \colon (\mathbb{F}_q[t]/Q\mathbb{F}_q[t])^\times \rightarrow \mathbb{C}^\times.
$$
As the number field case, we can define the Dirichlet $L$-function attached to $\varphi$ by
$$
L(T,\varphi) = \sum_{n = 0}^\infty b_{\varphi,n}T^n,
$$
where $b_{\varphi,n} = \sum_{f \in \mathcal{M}_n} \varphi(f)$, and $\mathcal{M}_n$ are the monics of degree $n$ in $\mathbb{F}_q[t]$.  It is not hard to show that if $\varphi$ is non-trivial then $L(T,\varphi)$ is a polynomial of degree at most $\deg Q -1$, since $b_{\varphi,n} = 0$ for $n \geq \deg Q$. 
In this context, we say $\varphi$ is primitive if there is no proper divisor $Q' \mid Q$ such that $\varphi(f) = 1$ whenever $f \equiv 1 \; \mathrm{mod} \; Q'$ and $(f,Q)=1$. I have often seen the claim that if $\varphi$ is primitive then $L(T,\varphi)$ has degree precisely $\deg Q-1$, which amounts to saying $b_{\varphi,\deg Q-1} \neq 0$. Unfortunately, after extensive search and several tries  I have not managed to find a proof of this fact. 
Whenever I see this it is usually stated in the same sentence as the Riemann hypothesis for curves over finite fields (Weil, 1948), which makes me wonder if this is a consequence of it. I doubt this however, since it seems like there should be an elementary argument.
I would  grateful for any suggestions, comments or ideas.
 A: For anyone who is curious about this in the future: I did find a proof in Corollary 2.4 of the paper 'The mean values of cubic $L$-functions over function fields' by David, Florea and Lalin (https://arxiv.org/abs/1901.00817).
The basic idea is to define for $f = \sum_{k \in \mathbb{Z}}\frac{f_k}{t^k} \in \mathbb{F}_q(t)$ a generalised exponential function
$$
 e_q(f) = \exp\left({\frac{2 \pi i \mathrm{Tr}(f_1)}{p}}\right),
$$
where $\mathrm{Tr} \colon \mathbb{F}_q \rightarrow \mathbb{F}_p$ is the usual trace function of field extensions. If we define the Gauss sum to be
$$
G(\varphi) = \sum_{f \; \mathrm{mod} \; Q} \varphi(f) e_q(f/Q),
$$
then like in the integer case it follows, that $G(\varphi) \neq 0$ for primitive $\varphi$. Finally, by rewriting $G(\varphi)$ it follows quite easily that
$$
G(\varphi) = \begin{cases}
      \tau(\varphi) b_{\varphi, \deg Q-1} \quad &\text{if } \varphi \text{ is odd}, \\
      -q \cdot b_{\varphi,\deg Q-1} &\text{if } \varphi \text{ is even}, 
     \end{cases}
$$
where $\tau(\varphi) = \sum_{a \in \mathbb{F}_q^\times} \varphi(a) e^{\frac{2 \pi i \mathrm{Tr}(a)}{p}} \neq 0$ if $\varphi$ is primitive.
There is probably a more 'standard' source for this, however it might be hard to find because the terminology is the same as in the number field case.
