On $D_{\mathrm{\ell }}$-extensions of odd prime degree $\mathrm{\ell }$

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{\mathrm{\ell }}$-extensions of odd prime degree $\mathrm{\ell }$ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics. We also obtain improved upper bounds for the number of such extensions (over Q) of bounded discriminant.