We prove that, for every modulus \({\mathfrak q}\) , every class of the narrow ray class group \(H_{\mathfrak q}({\textbf{K}})\) of an arbitrary number field \({\textbf{K}}\) contains a product of three unramified prime ideals \({\mathfrak {p}}\) of degree one with \({\mathfrak N}{\mathfrak {p}}\leqslant (t({\textbf{K}}){\mathfrak N}{\mathfrak q})^3\) , where \(t({\textbf{K}})\) is an explicit function of \({\textbf{K}}\) described in (1). 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 \(H_{\mathfrak q}({\textbf{K}})\) .
