N.J. Stat. § 19:27-6
In the case of a vacancy in the representation of this State in the United States Senate or House of Representatives, the writ may designate the next general election day for the election, but if a special day is designated, it shall specify the cause and purpose of such election, the name of the officer in whose office the vacancy has occurred, the day on which a special primary election shall be held, which shall be not less than 70 days nor more than 76 days following the date of such proclamation, and the day on which the special election shall be held, which shall be not less than 64 nor more than 70 days following the day of the special primary election. The writ shall also specify the day or days when the district boards shall meet for the purpose of making, revising or correcting the registers of voters to be used at such special election.
If the vacancy happens in the representation of this State in the United States Senate the election shall take place at the general election next succeeding the happening thereof, unless the vacancy shall happen within 70 days next preceding the primary election prior to the general election, in which case it shall be filled by election at the second succeeding election, unless the Governor shall deem it advisable to call a special election therefor, which he is authorized hereby to do.
If the vacancy happens in the representation of this State in the House of Representatives in any year, not later than the 70th day prior to the day for holding the next primary election for the general election, the Governor shall issue a writ of election to fill such vacancy, designating in said writ the next general election day as the day on which the election shall be held to fill such vacancy. The nomination of candidates to fill such vacancy shall be made in the same manner as the nomination of other candidates at the said primary election for the general election.
N.J.S. § 19:27-6