We study the Siegel-Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey-$s$, $s\>0$ category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey-$s$ formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin, for the analytic germ.