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 $[L:K]$? 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.

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.