  We study a one-dimensional Fermi gas in the presence of dissipative coupling
to environment through the Lindblad equation. The dissipation involves energy
exchange with the environment and favours the relaxation of electrons to
excitations. After switching on the dissipation, the system approaches a steady
state, which is described by a generalized Gibbs ensemble. The fermionic single
particle density matrix resembles deceivingly to that in a hermitian
interaction quench. It decays inversely with the distance for short times due
to the fermionic correlations in the initial state, which changes into a
non-integer power law decay for late times, representing dissipation induced
Luttinger liquid behaviour. However, the crossover between the two regions
occurs due to dissipation induced damping, and is unrelated to the propagation
of excitations. The velocity of information spreading is set by the dissipative
coupling, and differs significantly from the original sound velocity. The
thermodynamic entropy grows as $-t\ln t$ initially, and saturates to an
extensive value. Our results can be tested experimentally in one-dimensional
Dirac systems.
