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 ${\rm L}\big (\frac 12,\chi ^a\big ){\rm L}\big (\frac 12,\chi ^b\big )$ for arbitrary integers $a$ and $b$ when $\chi $ varies over Dirichlet characters of a given prime modulus.