My XS360 is a low number as well. I thinks its 1L9000323 or something like that. The transmission recently gave up on me and ruined the engine case.. I was not happy about it since the engine matched the frame...
Check voltage with the bike running and with it off. You have a mechanical voltage regulator from the factory. These are not known to be reliable and cause most of the charging issues on the 360s.. Batteries dont seem to last too long either. The mechanical regulators are adjustable but they are not precise at all. Even if your regulator is OK, Id still replace it with a solid state one.
As XSChris said, the hand controls are somewhat sensitive too. I had to take mine apart and clean them before all the buttons started working. Theres very small spring loaded parts in these so be careful if you take them down.