  Let $K$ be a number field with the discriminant $D_K$ and the class number
$h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove
that every ideal class of $K$ contains a prime ideal with norm less than
$h_{K}^2\log(D_K)^{2}$ and also all but $o(h_K)$ of them have a prime ideal
with norm less than $h_{K}\log(D_K)^{2+\epsilon}$. For imaginary quadratic
fields $K=\mathbb{Q}(\sqrt{D})$, by assuming Conjecture~\ref{piarcor} (a weak
version of the pair correlation conjecure), we improve our bounds by removing a
factor of $\log(D)$ from our bounds and show that these bounds are optimal.
