  We use the renormalization group method to study the normal state of
quasi-one-dimensional superconductors nearby a spin-density-wave instability.
On the basis of one-loop scattering amplitudes for the quasi-one-dimensional
electron gas, the integration of the renormalization group equations for the
two-loop single particle Matsubara self-energy leads to a nonFermi-liquid
temperature downturn of the momentum-resolved quasi-particle weight over most
part of the Fermi surface. The amplitude of the downturn correlates with the
entire instability line for superconductivity, defining an extended quantum
critical region of the phase diagram as a function of nesting deviations of the
Fermi surface. One also extracts the downward renormalization of interchain
hopping amplitudes at arbitrary low temperature in the normal phase. By means
of analytical continuation of the Matsubara self-energy, one-particle spectral
functions are obtained with respect to both energy and temperature and their
anomalous features analyzed in connection with the sequence of instability
lines of the phase diagram. The quasi-particle scattering rate is found to
develop an unusual temperature dependence, which is best described by the
superimposition of a linear and quadratic $T$ dependences. The nonFermi-liquid
linear-$T$ component correlates with the temperature scale $T_c$ of the
superconducting instability over an extended range of nesting deviations,
whereas its anisotropy along the Fermi surface is predicted to parallel the
momentum profile of a d-wave pairing gap on the Fermi surface. We examine the
implications of our results for low dimensional unconventional superconductors,
in particular the Bechgaard salts series of quasi-1D organic conductors, but
also the pnictide and cuprate superconductors where several common features are
observed.
