A variant of effective Chebotarev theorem

Question

Let $L$ be a Galois extension of a number field $K$ with the Galois group $G$. Let $N$ be the smallest integer with the following property: For any conjugacy class $C$ of $G$ there exists an unramified prime $\mathfrak{p}$ of $K$ such that $\text{Frob}_\mathfrak{p}\in C$ and $\text{Norm}_{K/\mathbb{Q}}\mathfrak{p}\le N$.

Is there any good bound of $N$, with respect to $|G|,[L:\mathbb{Q}]$ and $\text{disc}_{L/\mathbb{Q}}$? If $K=\mathbb{Q}$ and $L/K$ is abelian, this is done in Chapter 18 of this book. I'll be glad to know about the $K=\mathbb{Q}$ case.

Answer 1

There are bounds on $N$, but whether they are good or not I leave up to you to decide.

First, conditional on GRH, Lagarias and Odlyzko proved a bound on $N$ in their 1977 paper ''Effective versions of the Chebotarev density theorem''. This was made explicit by Bach and Sorenson in 1996 (in "Explicit bounds for primes in residue classes'' published in Mathematics of Computation). Their bound is $$ N \leq (4 \log({\rm disc}_{L/\mathbb{Q}}) + 2.5 [L : \mathbb{Q}] + 5)^{2}. $$

Unconditional results are substantially less good. In 2017, Zaman proved (in "Bounding the least prime ideal in the Chebotarev density theorem") that $N \ll ({\rm disc}_{L/\mathbb{Q}})^{40}$. A more complicated bound that is better in some situations was proven by Thorner and Zaman in their 2017 Algebra and Number Theory paper "An explicit bound for the least prime ideal in the Chebotarev density theorem".

To state Thorner and Zaman's result, choose an abelian subgroup $A$ of ${\rm Gal}(L/K)$ such that $A \cap C$ is nonempty, let $M = L^{A}$ be the fixed field of $A$ (so that $L/M$ is abelian). Let $Q$ be the maximum value of the norm (from $M$ to $\mathbb{Q}$) of the conductor of characters of $A$. Then $$ N \ll ({\rm disc}_{M/\mathbb{Q}})^{694} Q^{521} + ({\rm disc}_{M/\mathbb{Q}})^{232} Q^{367} [M : \mathbb{Q}]^{290 [M : \mathbb{Q}]}. $$