When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

Question

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which diverges). Now my question is can we classify set of all algebraic integers $\mathbb{\alpha} \in \mathbb{C}$ for which the sum $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[\alpha]$ ?