  Relativistic shocks are present in all high-energy astrophysical processes
involving relativistic plasma outflows interacting with their ambient medium.
While a well understood process in the context of relativistic hydrodynamics
and ideal magnetohydrodynamics, there is little to no understanding of their
propagation in media with a finite electrical conductivity. This work presents
a method for the derivation and solution of the jump conditions for
relativistic shocks propagating in MHD media with a finite electrical
conductivity. The covariant expressions of the jump conditions are derived and
the algebraic equations expressing the Rankine-Hugoniot conditions are solved
numerically. This method is employed for the solution of the Riemann problem
for the case of a forward and reverse shock which form during the interaction
of a gamma-ray burst ejecta with the circumburst medium in order to determine
the kinematics of the resulting blastwave and the dynamical conditions in its
interior. Our solutions clearly depict the impact of the plasma's electrical
conductivity in the properties of the post-shock medium. Two characteristic
regimes are identified with respect to the value of a dimensionless parameter
which has a linear dependence on the conductivity. For small values of this
parameter the shock affects only the hydrodynamic properties of the propagation
medium and leaves its electromagnetic field unaffected. No current layer forms
in the shock front; thus this is called the current-free regime. For large
values of this parameter the ideal MHD regime is retrieved. We also show that
the assumption of a finite electrical conductivity can lead to higher
efficiencies in the conversion of the ejecta energy into thermal energy of the
blastwave through the reverse shock. The theory developed in this work can be
applied to the construction of Riemann solvers for resistive relativistic MHD.
