Let $P_3=${$z\in Z[\sqrt{-3}],|z|^2 \text { is a prime number}, >3$}

Let $\alpha_1,...\alpha_n$ be distinct elements in $P_3$, and $l_1,...l_n\in Z^+$. Set $\alpha_1^{l_1}...\alpha_n^{l_n}=a+b\sqrt{-3}$.

Is it correct that $|\alpha_i|^2,a,b$ are mutual prime for all $i$?