Let $K$ be a number field. Let $P$ be a subset of the set of nonzero prime ideals in $K$. For $\mathfrak{p} \in P$, let $N(\mathfrak{p})$ be its absolute norm, so $N(\mathfrak{p}) = \#\mathcal{O}\_K/\mathfrak{p}$. The Dirichlet density of $P$ is defined as$$\lim\_{s \to 1^+} {{\sum\_{\mathfrak{p} \in P} N(\mathfrak{p})^{-s}}\over{\log {1\over{s-1}}}}.$$I have two questions.

1. What is the Dirichlet density of the set of all prime ideals?
2. Let $P$ be the set of prime ideals in $\mathbb{Q}(i)$ that lie above primes congruent to $3$ mod $4$. What is the Dirichlet density of $P$?