In a notes I found the following exercise and solution:
I have a question. In the proof I admit the statement "the Dirichlet density of these prime ideals is $1/2$ " but i do not understand why the author says " taking into account that there are two ideals of norm $p$ in the principal class gives us the final density of $1/4$. I think that there are also principal ideals $(a+b\sqrt{-5})$ whose norm is $p^2$. Does those principal ideals have dirichlet density $0$?