  The remnant radio galaxies in galaxy clusters are important sources of seed
relativistic electron population in the intra-cluster medium (ICM). Their
occurrence and spectral properties are poorly studied. In this work we present
a broadband study of the radio relic in the galaxy cluster Abell 4038 using the
Upgraded Giant Metrewave Radio Telescope (uGMRT). We present the uGMRT images
in the bands 300 - 500 MHz and 1050 - 1450 MHz having rms noise $70\,\mu$Jy
beam$^{-1}$ and $30\,\mu$Jy beam$^{-1}$, respectively, that are the deepest
images of this field so far. A spectral analysis of the relic over 300 - 1450
MHz using images in sub-bands scaled to have constant fractional bandwidths to
achieve a closely matched uv-coverage was carried out. The 100 kpc extent of
the relic is divided into Loop, Arc, Bridge and North-end. The Loop has a steep
spectral index of $\alpha=2.3\pm0.2$ ($S_{\nu}\propto\nu^{-\alpha}$). The
North-end has ultra-steep spectra in the range $2.4 - 3.7$. The Arc is found to
skirt a curved region seen in the \emph{Chandra} X-ray surface brightness image
and the highest spectral curvature in it reaches $1.6\pm0.3$. We interpret the
morphology and spectral properties of the relic in the scenario of an
adiabatically compressed cocoon from the past activity of the Brightest Cluster
Galaxy in the cluster. A comparison of the properties of the A4038 relic with a
sample of 10 such relics is discussed.
