The Lottery Headquarters is different for every State. For example, The Lottery Company's Headquarters in Conneticut is located in Rocky Hill, CT. The Address to The Lottery Company is 777 Brook Street, Rocky Hill, CT 06067.
The headquarters for the Illinois DMV is located at 213 State Capitol, Springfield, Illinois 62756. This is the same as the Departments of the Illinois Secretary of State (Jesse White), which is the parent agency of Illinois DMV.