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 $\chi$. Several authors have explored this phenomenon for Dirichlet characters $\chi$ with fixed modulus and having the same parity as $k$. We extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers across arbitrary number fields. In the process, we determine the dimension of the multi-dimensional generalization of cotangent values and the sum of generalized Chowla-Milnor spaces over the linearly disjoint number fields.