  We prove that, for every modulus $\mathfrak{q}$, every class of the narrow
ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field
$\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$
of degree one with $\mathfrak{N}\mathfrak{p}\le
(t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an
explicit function of $\mathbf{K}$ described in the paper. To achieve this
result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray
classes and then an equally explicit improved Brun-Titchmarsh Theorem for large
subgroups of narrow ray class groups. En route, we deduce an explicit upper
bound for the least prime ideal in a quadratic subgroup of a narrow ray class
group and also for the size of the least ideal that is a product of degree one
primes in any given class of $H_\mathfrak{q}(\mathbf{K})$.
