  We compute non-perturbatively the renormalization constants of composite
operators on a quenched $16^3 \times 28 $ lattice with lattice spacing $a$ =
0.20 fm for the overlap fermion by using the regularization independent (RI)
scheme. The quenched gauge configurations were generated with the Iwasaki
action. We test the relations $Z_A = Z_V$ and $ Z_S=Z_P$ and find that they
agree well {(less than 1%)} above $\mu$ = 1.6 GeV. %even for our lattice with a
coarse lattice spacing. We also perform a Renormalization Group (RG) analysis
at the next-to-next-to-leading order and match the renormalization constants to
the $\bar{\rm MS}$ scheme. The wave-function renormalization $Z_{\psi}$ is
determined from the vertex function of the axial current and $Z_A$ from the
chiral Ward identity. Finally, we examine the finite quark mass behavior for
the renormalization factors of the quark bilinear operators. We find that the
$(pa)^2$ errors of the vertex functions are small and the quark mass dependence
of the renormalization factors to be quite weak.
