On the special values of automorphic L-functions

Let π be a cuspidal representation of $\text{GL}{}_{\mathrm{n}}\text{(}\text{A}{}_{\mathrm{<mtext>Q</mtext>}}\text{)}$ with non-vanishing cohomology and denote by L(π,s) its L-function. Under a certain local non-vanishing assumption, we prove the rationality of the values of L(π?χ,0) for characters χ, which are critical for π. Note that conjecturally any motivic L-function should coincide with an automorphic L-function on GL_n; hence, our result corresponds to a conjecture of Deligne for motivic L-functions. To cite this article: J. Mahnkopf, C. R. Acad. Sci. Paris, Ser. I 338 (2004).