  To detect off-shell Abelian $Z'$ boson in $e^+e^-\to\mu^+\mu^-$, we propose
one-parameter observables with the best value-to-uncertainty ratio. The
observables are constructed by angular integration of the differential cross
section with smooth weight functions. The value-to-uncertainty ratio is used as
a criterion to select the unique weight function leading to the observable with
the best statistical power for data analysis. The observables allow to select
either vector or axial-vector $Z'$ couplings to leptons. The obtained
observable can be useful in future experiments at lepton colliders such as the
ILC.
