Linear independence of values of Dirichlet $L$ functions

In this paper, for a given Dirichlet character mod $N$ with $4\nmid N$, we give a lower bound of order $\sqrt{s\mathrm{/}\log (s)}$ for the dimension of the $\mathbb{Q}(e^{2i\pi \mathrm{/}N})$-vector space spanned by the values of its $L$-function at integers $\le s$ of a given parity. We thus generalize a result Fischler proved in 2021, corresponding to the principal character mod 1. To this end, we construct linear combinations of these values of $L$-function with a refined version of Siegel’s lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on a ``Shidlovskii’s lemma’'.