  We study a holographic superconductor model with momentum relaxation due to
massless scalar fields linear to spatial coordinates($\psi_I = \beta
\delta_{Ii} x^i$), where $\beta$ is the strength of momentum relaxation. In
addition to the original superconductor induced by the chemical
potential($\mu$) at $\beta=0$, there exists a new type of superconductor
induced by $\beta$ even at $\mu=0$. It may imply a new `pairing' mechanism of
particles and antiparticles interacting with $\beta$, which may be interpreted
as `impurity'. Two parameters $\mu$ and $\beta$ compete in forming a
superconducting phase. As a result, the critical temperature behaves
differently depending on $\beta/\mu$. It decreases when $\beta/\mu$ is small
and increases when $\beta/\mu$ is large, which is a novel feature compared to
other models. After analysing ground states and phase diagrams for various
$\beta/\mu$, we study optical electric($\sigma$), thermoelectric($\alpha$), and
thermal($\bar{\kappa}$) conductivities. When the system undergoes a phase
transition from a normal to a superconducting phase, $1/\omega$ pole appears in
the imaginary part of the electric conductivity, implying infinite DC
conductivity. If $\beta/\mu <1$, at small $\omega$, a two-fluid model with an
imaginary $1/\omega$ pole and the Drude peak works for $\sigma$, $\alpha$, and
$\bar{\kappa}$, but if $\beta/\mu >1$ a non-Drude peak replaces the Drude peak.
It is consistent with the coherent/incoherent metal transition in its metal
phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases
even when $\mu=0$.
