Any qualified elector of any senatorial district, as defined in Section 5-1-3, who possesses the necessary constitutional qualifications may qualify as a candidate and be elected a state senator without regard to any so-called gentleman's agreement or any other restriction. All elections for the office of state senator shall be conducted on a senatorial district basis regardless of the number of counties composing said district, and the names of qualified candidates therefor shall be placed on the ballots in each precinct of every county within said senatorial district.