Fan clutch is likely faulty.
Bmw e46 m3 fan clutch test.
With your vehicle info handy give the pelican parts parts specialists a call at 1 888 280 7799.
They can figure out what part or repair kit you need.
Followup from the pelican staff.
Somewhere around the middle of temp gauge i have a 2004 bmw 325ci e46.
Open hood spin the fan by hand there should be a slight resistance and slowing it down to a dead stop.
Car must be turned off don t try to stop it while running.