I don't think it has anything to do with 12 volts, as the sender doesn't use 12 volts to send a signal back to the gauge. I think you will need to check it with an ohmmeter, but I couldn't tell you what it should read if it's good.
You probably can hook the ohmmeter up to the sender when cold and take a reading. Then warm it up, either in the engine or a pan of water, and see if the reading changes. But if this is the correct way of checking it, I have no idea what the readings would be.
Just an uneducated guess.
