Thanks a lot. That's what I thought as well. I think the previous owner put an HID kit on the bike, but only put a single output H4 bulb in, so it only comes on when I turn the high beams on but there is no low beam/driving lamp on when the motor is running. This is what's in there now:http://moonlightnings.com/wp-content/uploads/2009/01/h4-single-xenon-300x201.jpg I guess I need to swap that out for a H4 h/l bulb. Thanks for your help. I appreciate it a lot.
The headlamp on the naked FZ6 utilize a h4 bulb, in other words you got both the driving light and the high beam in one single bulb. The other lamp you're talking about, is the parking lamp. That is supposed to be on at all times as long as the ignition is turned on. My driving lamp don't turn on unless the engine is running, and obviously the high beam don't go on until I switch on the high beam.
It seems to me that your parking light bulb is done, and you need to buy a new one.