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The origin of neutrino mass is a big unsolved problem of the Standard Model (SM) that motivate us to consider beyond the SM (BSM) scenarios where SM-singlet right-handed neutrinos (RHNs) are introduced to explain the origin of the light neutrino masses through the seesaw mechanism. There is a variety of ways which could lead us to this goal and one of them is a general U$(1)$ extension of the SM. In this scenario, three SM-singlet RHNs are introduced to cancel the gauge and mixed gauge gravity anomalies. After anomaly cancellation, we notice that the left- and right-handed charged fermions are differently charged under the general U$(1)$ gauge group evolving a chiral scenario. After the breaking of the general U$(1)$ symmetry, a neutral BSM gauge boson $(Z^\prime)$ acquires mass and it is a free parameter. Such $Z^\prime$, being lighter than $5$ GeV, could be probed at the intensity and lifetime frontiers like FASER, FASER2, DUNE, and ILC beam dump experiments. The estimated bounds are needed to be compared with the existing bounds. We find that existing constraints from Orsay, Nomad, PS191, KEK, LSND, CHARM experiments, and cosmological scenario like SN1987A can be compared in our case once estimated for chiral scenarios. Finally, we compare the parameter spaces showing viable ones that could be probed by FASER, FASER2, DUNE, and ILC beam dump experiments and already excluded regions from Orsay, Nomad, PS191, KEK, LSND, CHARM, and SN1987A for a chiral scenario.