running light

i have a 1977kz1000a1 and just wanted to know if tail ligth should shine when bike is running and no brakes applied ?

Yep

It should be on dim if the headlight is on, it has a dual filament bulb, one for the tail light the other for the brake light. Press on the rear brake pedal and see if it gets brighter. If it doesn't one of the switches is not releasing. It's usually the rear switch spray into it with wd40 and work it a few times. If you have a manual check the adjustment.

As stated above, it should when the headlight is on. The 77 US model does not have on-all-time lights though, so unlike a newer bike it will not be on when the headlight is off (an option you don't have on a newer bike). If it is not on, and the bulb is good, socket is clean and good, most obvious place to look is where the wiring has disconnects to pull it through the hole in the fender when removing the tail light assembly. Just follow the wiring until you find the little "bullet" connectors. It can also come un-hooked at that big mess of connectors under the gas tank

Ron