  We show how to orient moduli spaces of holomorphic disks with boundary on an
exact Lagrangian immersion of a spin manifold into complex n-space in a
coherent manner. This allows us to lift the coefficients of the contact
homology of Legendrian spin submanifolds of standard contact (2n+1)-space from
Z_2 to Z. We demonstrate how the Z-lift provides a more refined invariant of
Legendrian isotopy. We also apply contact homology to produce lower bounds on
double points of certain exact Lagrangian immersions into C^n and again
including orientations strengthens the results. More precisely, we prove that
the number of double points of an exact Lagrangian immersion of a closed
manifold M whose associated Legendrian embedding has good DGA is at least half
of the dimension of the homology of M with coefficients in an arbitrary field
if M is spin and in Z_2 otherwise.
