My temp gauge reads overheat after starting the engine without overheating. but the ecm gets the correct information and switch on the fan at a correct moment. does my sensor or gauge bad

sure sounds like your temp sensor is bad.it sends a signal to the ECM to turn the fan(s) on and tells the gauge what the temp is.Part Number: SU5541...$14.99 at the ZONE

Does your car use a separate temp sensor for the gauge?

Temp sensor needed