For every finite extension $K$ of $\mathbb{Q}$, for every finite Galois extension $L$ of $K$ and every conjugacy class $\mathcal{C}$ of Gal$(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ that is unramified in $L$ and
$$ \bigg[ \frac{L/K}{\mathfrak{p}} \bigg] = \mathcal{C}$$ as wella as $$ N_{K/\mathbb{Q}}(\mathfrak{p})$$ is a rational prime satisfying $$N_{K/\mathbb{Q}}(\mathfrak{p}) \leq |d_{L}|^{12577}\hspace{2mm}.$$
This is the state of the art bound for the least prime ideal in the Chebotarev density theorem I found in this paper. There are works that prove that under GRH one can improve the upper bound $|d_{L}|^{12577}$ above to $c \cdot log|d_{L}|$ for some constant $c$.
The Riemann hypothesis for function fields $\mathbb{F}_{q}(T)$, where $q$ is some prime power, is already known. However, I could not find equivalent result about least prime ideal in the Chebotarev density theorem in function fields, despite wide searching.
Has the analogous result already been done or is too trivial in the latter case? If already done, can anyone explain the result and/or point me in the right direction.