Q: after sitting for a week fuel gauge not working sitting on E

This is somewhat common on the later model Corvettes. Try unhooking the battery to see if a "global reset" will correct this as sometimes this allows the controllers to reset. The Corvette (as well as many other vehicles) has an on-board network where many of the car’s systems exchange information across a local network (in this case GMLAN), which works much like many of today’s computer networks in an office setting.

There are many sensors on-board the vehicle which send input data to these networks, which then translate information across the network in the vehicle. Occasionally, what can happen is these systems (much like today’s computer networks) can freeze up due to an excessive amount of network "traffic" causing one component or another to not work properly temporarily. Unhooking the battery and "restarting the system" sometimes can correct this. If this does not work, I would recommend taking it to the Chevrolet dealership to have it reset.