  We explore a holographic superconductor model in which a real scalar field is
non-minimally coupled to a gauge field. We consider several types of the
non-minimal coupling function h($\psi$) including exponential, hyperbolic
(cosh), power-law and fractional forms. We investigate the influences of the
non-minimal coupling parameter $\alpha$ on condensation, critical temperature
and conductivity. We can categorize our results in two groups. In the first
group, conductor/superconductor phase transition is easier to occur for larger
values of $\alpha$, while in the second group stronger effects of the
non-minimal coupling makes the formation of scalar hair harder. Although the
real and imaginary parts of conductivity are impressed by different forms of
h($\psi$), they follow some universal behaviors such as connecting with each
other through Kramers-Kronig relation in low frequency regime or the appearance
of gap frequency at low temperatures. We find the best form of forms of
non-minimal coupling function that gives us better information in wide range of
non-minimal coupling constant and temperature. Choosing the best form of
h($\psi$), we construct a family of solutions for holographic
conductor/superconductor phase transitions to discover the effect of the
hyperscaling violation when the gauge and scalar fields are non-minimally
coupled. we find that the critical temperature increases for higher effects of
hyperscaling violation $\theta$ and non-minimal coupling constant $\alpha$. By
increasing these two parameters, we obtain lower values of condensation which
means that conductor/superconductor phase transition will acquire easier.
Furthermore, we understand that the hyperscaling violation affects the
conductivity $\sigma$ of the holographic superconductors and changes the
expected relation in the gap frequency. Some universal behaviors like infinite
DC conductivity are observed.
