  For a number field $K$, the Euler-Kronecker constant $\gamma_K$ associated to
$K$ is an arithmetic invariant the size and nature of which is linked to some
of the deepest questions in number theory. This theme was given impetus by
Ihara who obtained bounds, both unconditional as well as under GRH for Dedekind
zeta functions. In this note, we study the analogous constants associated to
the narrow ray class fields of an imaginary quadratic field. Our goal for
studying such families is twofold. First to show that for such families, the
conditional bounds obtained by Ihara can be improved on the average, again
under GRH for Dedekind zeta functions. Further, our family of number fields are
non-abelian while such average bounds have earlier been studied for cyclotomic
fields. The technical part of our work is to make the dependence of these upper
bounds on the ambient number field explicit. Such explicit dependence is
essential to further our objective.
