A 2012, if it is an L2, would normally be a glee.
The original 1,000 was christened ( imaginatively ) a vee.
The original 650, being smaller, became a wee.
The L2 used the engine from the Gladius, so became GLadius engined wEE, or glee.
After that, Suzuki put out a new 1,000, with a beak. So the Veek was born. Glees then got a beak, and became gleeks. It got silly after that with the single headlight and newer models.
Stick a photo up to make sure it's not an old stock wee registered in 2012.