DING DING DING!!! We have a winner. It's the fuel pump. It pressurizes the system prior to starting the car so it starts faster and easier. Eventually when the car is off the pressure will bleed away through the regulator back to the tank (in the front) so it re-pressurizes when you turn on the key. Most cars do this (very few dont).
It should only last a few seconds, not like a minute or anything tho...
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