Memory (RAM) of computers after shut down
It's not like a powerdown deliberately erases data, so it's mainly leakage currents through which a bit's value is lost over time. That can be minutes, but you can't rely on that; it may also be seconds, and depending on your data, a couple of changed bits can render all of it worthless, for instance if they're looking for encryption keys.  

The police will have a hard time with it. It will have to remove the memory, because powering the PC on again will overwrite much of it, place the memory in another computer which will not do so, and do all of this as quickly as possible. I don't think they do this.  

It should be noted that low temperatures will decrease leakage current from the memory cell's capacitor, so to a certain extent you can "freeze" memory.