A closed formula for the density in Artin's primitive root conjecture over number fields

Let $K$ be a number field and $\alpha\in K^\times$. Artin's primitive root conjecture concerns the proportion of prime ideals $\mathfrak p$ of $K$ for which $(\alpha \bmod \mathfrak p)$ is well-defined, non-zero, and generates the multiplicative group at $\mathfrak p$. The conjecture states that the set of those prime ideals admits a natural density and provides a formula for it.
In 1967, assuming GRH, Hooley proved Artin's conjecture (for the integers) over $\mathbb Q$. He has also given a closed formula for the density, describing the rational number that is the ratio between the density and the Artin constant. In 1975 Cooke and Weinberger proved the analogue result over number fields, and we now generalize Hooley's closed formula to all number fields.