Except as otherwise provided by Section 23-15-961, a person desiring to contest the election of another person returned as the nominee of the party to any county or county district office, or as the nominee of a legislative district composed of one (1) county or less, may, within twenty (20) days after the primary election, file a petition with the secretary, or any member of the county executive committee in the county in which the election was held, setting forth the grounds upon which the primary election is contested; and it shall be the duty of the executive committee to assemble by call of the chairman or three (3) members of said committee, notice of which contest shall be served five (5) days before said meeting, and after notifying all parties concerned proceed to investigate the grounds upon which the election is contested and, by majority vote of members present, declare the true results of such primary.