Except as otherwise provided in Section 23-15-961, a person desiring to contest the election of another returned as the nominee in state, congressional and judicial districts, and in legislative districts composed of more than one (1) county or parts of more than one (1) county, upon complaint filed with the Chairman of the State Executive Committee, by petition, reciting the grounds upon which the election is contested. If necessary and with the advice of four (4) members of said committee, the chairman shall issue his fiat to the chairman of the appropriate county executive committee, and in like manner as in the county office, the county committee shall investigate the complaint and return their findings to the chairman of the state committee. The State Executive Committee by majority vote of members present shall declare the true results of such primary.