On separation of quadratic forms on the imaginary quadratic field in its Hilbert class field

Let $K^{(1)}$ be the Hilbert class field of the imaginary quadratic field $K=Q(\sqrt {d}),d<0.$ We derive the product representations of a class of Dirichlet L-series associated with the character group constructed from the Artin map between the ideal class group of $K$ and the Galois group $Gal(K^{(1)}/K)$. The application of the Mellin transform to the product representations of these Dirichlet series yields a family of generating functions for representations of positive integers by the subgroups of the quadratic forms.