  We study three properties of a holographic superconductor related to
conductivities, where momentum relaxation plays an important role. First, we
find that there are constraints between electric, thermoelectric and thermal
conductivities. The constraints are analytically derived by the Ward identities
regarding diffeomorphism from field theory perspective. We confirm them by
numerically computing all two-point functions from holographic perspective.
Second, we investigate Homes' law and Uemura's law for various high-temperature
and conventional superconductors. They are empirical and (material independent)
universal relations between the superfluid density at zero temperature, the
transition temperature, and the electric DC conductivity right above the
transition temperature. In our model, it turns out that the Homes' law does not
hold but the Uemura's law holds at small momentum relaxation related to
coherent metal regime. Third, we explicitly show that the DC electric
conductivity is finite for a neutral scalar instability while it is infinite
for a complex scalar instability. This shows that the neutral scalar
instability has nothing to do with superconductivity as expected.
