It does, and typically it's at lower voltage than 4,000 rpm and up. However, rather than start buying stuff and throwing it at the bike, go to Radio Shack, etc, and buy multimeter and some simple tests. You need a fresh battery. Charge it and let it sit overnight. Should be 12.6 or so volts. If not, it's not fresh. If you charge this current battery overnight, let it SIT for at least an hour, should be 12.6 or so. If it's less, replace it.
Then install the fresh battery, start the motor, and check voltage at idle (should be at least 13.xx) and at 4,000 rpm, where it should be 14+ volts DC, but on these GSXR's it will often be 15 volts. If it's under 14, the R/R is probably bad. BUT, before you buy an RR, unplug the stator cable, start the bike and check the stator output voltage from pin to pin, A to B, A to C, and B to C. All three should be over sixty volts at 4,000 rpm ( AC, not DC) If they are, you need to replace the R/R. Don't install OEM, it's shit, get a Rick's.
So, first get fresh battery, test output voltage at battery, then output voltage from stator.
As to charging at idle, as mentioned, the AGM battery likes 14.4 volts. What happens when you "charge" at idle is that you typically get 13.X volts, which is theoretically enough to make everything work, but it doesn't. The battery will not charge fully until you get charging voltage to over 14.