  We have studied in detail the 0.15-15 GHz radio spectrum of the gamma-ray
binary LS 5039 to look for a possible turnover and absorption mechanisms at low
frequencies, and to constrain the physical properties of its emission. We have
analysed two archival VLA monitorings, all the available archival GMRT data and
a coordinated quasi-simultaneous observational campaign conducted in 2013 with
GMRT and WSRT. The data show that the radio emission of LS 5039 is persistent
on day, week and year timescales, with a variability $\lesssim 25~\%$ at all
frequencies, and no signature of orbital modulation. The obtained spectra
reveal a power-law shape with a curvature below 5 GHz and a turnover at
$\sim0.5$ GHz, which can be reproduced by a one-zone model with synchrotron
self-absorption plus Razin effect. We obtain a coherent picture for a size of
the emitting region of $\sim0.85~\mathrm{mas}$, setting a magnetic field of
$B\sim20~\mathrm{mG}$, an electron density of $n_{\rm e}\sim4\times10^5~{\rm
cm^{-3}}$ and a mass-loss rate of $\dot M\sim5\times10^{-8}~{\rm M_{\odot}
yr^{-1}}$. These values imply a significant mixing of the stellar wind with the
relativistic plasma outflow from the compact companion. At particular epochs
the Razin effect is negligible, implying changes in the injection and the
electron density or magnetic field. The Razin effect is reported for first time
in a gamma-ray binary, giving further support to the young non-accreting pulsar
scenario.
