Abstract: This paper presents a search for a new Z' vector gauge boson with the ATLAS experiment at the Large Hadron Collider using pp collision data collected at root s = 13TeV, corresponding to an integrated luminosity of 139 fb(-1). The new gauge boson Z' is predicted by L-mu – L-tau models to address observed phenomena that can not be explained by the Standard Model. The search examines the four-muon (4 mu) final state, using a deep learning neural network classifier to separate the Z' signal from the Standard Model background events. The di-muon invariant masses in the 4 μevents are used to extract the Z' resonance signature. No significant excess of events is observed over the predicted background. Upper limits at a 95% confidence level on the Z' production cross-section times the decay branching fraction of pp -> Z' μμ-> 4 μare set from 0.31 to 4.3 fb for the Z' mass ranging from 5 to 81 GeV. The corresponding common coupling strengths, g(Z'), of the Z' boson to the second and third generation leptons above 0.003 – 0.2 have been excluded.