  Among the $Z$-pole observables, $A_{FB}^{(0,b)}$ and $A_e$ suffer
moderately-large standard deviations from the Standard Model predictions.
Fine-tuning of the unknown Higgs mass only reduces the deviation of one of them
at the expense of increasing the deviation of the other observable. If we take
this fact seriously, the result can be interpreted as independent experimental
evidence of existing new physics beyond SM, even if a 125GeV Higgs on the LHC
is finally confirmed to be SM Higgs. We show in this paper that the existence
of a $Z'$ boson mixing with $Z$ and $\gamma$ and with generation-dependent
anomaly-free charge assignments, helps to suppress $A_{FB}^{(0,b)}$ and $A_e$
at $Z$-pole simultaneously and dose reduce the largest deviation from
$2.7\sigma$ in SM predictions to $1.2\sigma$ in our scenario. The global
electroweak fit does not prefer $Z'$ coupling to the second-generation quarks
and the first-generation left-handed lepton and right-handed electron. The
fitting result also supports a negative contribution to the $S$ parameter from
the $Z'$ boson.
