  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
$L(\demi,\chi^a)L(\demi,\chi^b)$ for arbitrary integers~$a$ and~$b$ when $\chi$
varies over Dirichlet characters to a prime modulus.
