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We present a high statistic systematic study of the overlap correlationfunction well below the critical temperature in the three dimensional Gaussianspin glass. The off-equilibrium correlation function has been studiedconfirming the power law behavior for the dynamical correlation length. Inparticular we have computed the dynamical critical exponent $z$ in a wide rangeof temperatures, $0.35 \le T \le 0.9$, obtaining a dependence $z(T)=6.2/T$ in avery good agreement with recent experiments. Moreover, we report a study of theviolation of the fluctuation-dissipation theorem for very low temperatures$T=0.5$ and $T=0.35$. All our numerical results avoid a droplet modelinterpretation even when $T$ is as low as $T=0.35$.Comment: LaTeX, 14 pages and 5 figures. A minor arithmetic error corrected and  references update

Off-equilibrium dynamics at very low temperatures in three-dimensional spin glasses