  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
$\sqrt{s} = 13$ TeV, 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 $4\mu$ events 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 $pp
\rightarrow Z'\mu\mu \rightarrow 4\mu$ are 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.
