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 $alpha=2.3pm0.2$ ($S_{ u}propto u^{-alpha}$). 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 $1.6pm0.3$. 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.
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