We investigate the physical nature and origin of the gamma-ray emission from the extended source HESS J1841-055 observed at TeV and GeV energies. We observed HESS J1841-055 at TeV energies for a total effective time of 43 hours with the MAGIC telescopes, in 2012 and 2013. Additionally, we analysed the GeV counterpart making use of about 10 years of Fermi-LAT data. Using both Fermi-LAT and MAGIC, we study both the spectral and energy-dependent morphology of the source for almost four decades of energy. The origin of the gamma-ray emission from this region is investigated using multi-waveband information on sources present in this region, suggested to be associated with this unidentified gamma-ray source. We find that the extended emission at GeV-TeV energies is best described by more than one source model. We also perform the first energy-dependent analysis of the HESS J1841-055 region at GeV-TeV. We find that the emission at lower energies comes from a diffuse or extended component, while the major contribution of gamma rays above 1 TeV arises from the southern part of the source. Moreover, we find that a significant curvature is present in the combined observed spectrum of MAGIC and Fermi-LAT. The first multi-wavelength spectral energy distribution of this unidentified source shows that the emission at GeV-TeV energies can be well explained with both leptonic and hadronic models. For the leptonic scenario, bremsstrahlung is the dominant emission compared to inverse Compton. On the other hand, for the hadronic model, gamma-ray resulting from the decay of neutral pions ($pi^0$) can explain the observed spectrum. The presence of dense molecular clouds overlapping with HESS J1841-055 makes both bremsstrahlung and $pi^0$-decay processes the dominant emission mechanisms for the source.
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