  While the zeta function does not determine a number field uniquely, the
$L$-series of a well-chosen Dirichlet character does. Moreover, isomorphisms
between two number fields are in natural bijection with $L$-series preserving
isomorphisms of $l$-torsion subgroups of the Dirichlet character groups. We
extend this by showing that homomorphisms between number fields are in natural
bijection with group homomorphisms between $l$-torsion subgroups of the
Dirichlet character groups abiding a divisibility condition on the $L$-series
when $l$ is sufficiently large.
