What area in CA do you live?
How old are you?
Yes, you are supposed to ride in the bike lane.............and
Motorized bicycles in CA are allowed 2 hp or less and not supposed to go over 30 mph.
There is no engine cc limit and in Jan 2015 the law will be updated to 4 hp or less.
You also need a drivers license with an M2 or M1.
Wear a DOT helmet.
Get the $20 lifetime motorized bike license plate from the DMV.
Left side mirror, horn, lights.
Thats the main points even though these laws may not be enforced in a lot of areas in CA.