Does the private key have special mathematical properties?
In Bitcoin, a private key is just a random number between 1 and 0xFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141. This number is derived from the order of the elliptic curve that Bitcoin uses, secp256k1. 

Note that you don't have to have the most significant bit 2^255 set to be a valid Bitcoin private key, any number greater than or equal than 1 will work. Indeed, knowing that the most significant bit is set actually lessens the security of that private key, because it reveals some information about the key. You don't have to use some special key generator, in fact there are even tutorials on how to generate a private key with coin flips. The only requirement of a private key (other than being in the range outlined above), is that it be random (hard to guess).