What are the conditions for locating the specific location of the mobile phone number in the public security bureau? How to apply?

You don't need to apply for this kind of thing yourself. The public security organ decides whether to take such measures according to the circumstances of the case, which is generally done by the Eighth Division (Technical Investigation Division) of the Municipal Bureau.

Generally, this method should be used in major cases, or cases with great and bad influence, or gang cases, because the cost is very high.

It's relatively simple to just retrieve the phone bill, SMS, or the location of the base station, but it's also decided by the public security bureau itself and has nothing to do with the application.