My Mazda 6 won't work the radiator fan at any temp (like it dead)

My Mazda 6 (2003 model with 2.0L gasoline engine) won't work the
radiator fan at any high temperature. But, both fans are running whenever
the A/C is On. But radiator fan doesn't work alone without A/C.
Temperature indicator (Heat indicator) also not over-passing the middle
position and normally at in the middle when the engine came to normal
operating temperature. I have checked the temperature sensor by
removing the socket (socket with two wires) and suddenly, both fans
started to run at very high speed continuously and engine warning lamp
came. After connecting the socket back, both fans stopped and system
came back to normal.