  The study of linear independence of $L(k, \chi)$ for a fixed integer $k>1$
and varying $\chi$ depends critically on the parity of $k$ vis-\`a-vis $\chi$.
This has been investigated by a number of authors for Dirichlet characters
$\chi$ of a fixed modulus and having the same parity as $k$.The focal point of
this article is to extend this investigation to families of Dirichlet
characters modulo distinct pairwise co-prime natural numbers. The interplay
between the resulting ambient number fields brings in new technical issues and
complications hitherto absent in the context of a fixed modulus (consequently a
single number field lurking in the background). This entails a very careful and
hands-on dealing with the arithmetic of compositum of number fields which we
undertake in this work. Our results extend earlier works of the first author
with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata.
