First of all, welcome to the forum.
The 1200s were well known for stator failures, the internal alternator, if you will. If you check the voltage at the battery, with it running at 2000 RPM, you should have ~13.5-14 Volts, if not, the stator isn't functioning. If that is the case, it could cost you lots of time and money to repair.
2004 GL1800 Bloodstone Red Metallic
1988 GL1500/6 Martini Beige Metallic (Sold)
1984 GL1200 Pearl Saturn Red/Century Brown Metallic (Sold)
1981 Honda CM400T Candy Sword Blue with Metallic Light Blue (Cubed and Recycled)
Homo praesumitur bonus donec probetur malus