13.1V will still charge the battery. Maybe not to 100%, but it will sure be higher than the 11.7V you're getting out of it. When the engine is revved, you should be at 13.5V or higher. With a garbage battery installed, the alternator is going to be under some major strain trying to charge it back up. At idle, you're well below the rated amperage capacity as well. Those two things together add to up to a less than ideal charging voltage.

Get a new battery. If the one you have isn't very old, claim it under warranty.

__________________
Current stable:
1995 E320 149K (Nancy)
1983 500SL 120K (SLoL)

Black Sheep:
1985 524TD 167K (TotalDumpster™)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)