Toroidal families and averages of L-functions, I

We initiate the study of certain families of $L$-functions attached to characters of subgroups of higher-rank tori, and of their average at the central point. In particular, we evaluate the average of the values $L(\text{demi},{\chi }^a)L(\text{demi},{\chi }^b)$ for arbitrary integers~$a$ and~$b$ when $\chi$ varies over Dirichlet characters to a prime modulus.