The controller shall examine, adjust, and determine all accounts and claims against the state, for the payment of which provision shall have been made by law, and by appropriations made therefor, and shall certify to the general treasurer the amount due and allowed by him or her on every account or claim, the head of expenditures to which the same is to be charged, and the law authorizing the payment thereof, and the law making appropriation therefor; and all accounts so certified shall, for each year, be regularly recorded by the state controller.