  We compute the ratio between the scale $\Lambda_L$ associated with a lattice
formulation of QCD using the overlap-Dirac operator, and $\Lambda_{MS-bar}$. To
this end, the one-loop relation between the lattice coupling $g_0$ and the
coupling renormalized in the MS-bar scheme is calculated, using the lattice
background field technique.
  We also compute the one-loop renormalization $Z_\Gamma$ of the two-quark
operators $\bar{\psi} \Gamma \psi$, where $\Gamma$ denotes a generic Dirac
matrix. Furthermore, we study the renormalization of quark bilinears which are
more extended and have better chiral properties.
  Finally, we present improved estimates of $Z_\Gamma$, coming from cactus
resummation and from mean field perturbation theory.
