I don't know. All I know is I've seen cop cars locked when answering a call but they are running, and they will be on for the 20-30 mins or more that the cops are doing some noise complaint or domestic violence

Depends. Generally they are all powered by the vehicle's battery. When turned off some remain powered. This slowly drains the battery. It's why some vehicle's batteries will die after extended periods without starting and charging.

However, many are on the other side of a relay. When you start your vehicle a relay turns on and this initiates power to various control modules (computers). For example, your transmission control module doesn't need to be powered while the vehicle is turned off. Whereas your body control module does (aux lights, radio, door locks, windows, and so on).

That said, vehicle power is notoriously dirty. ICs hate dirty power. Especially during start up and shutdown. Even including several seconds after initial start as other modules power up and relays continue to close their contacts, increasing power draw and noise throughout.

Start up and shutdown is very hard on computers. This is why most electronics used in vehicles have specific ratings for such environments. For both required temp range and the electrical hostility to the components connected.