The RealSense SDK does not usually require a product serial number. The only serial number involved is one that is given to you when you register your personal details before downloading the SDK. This link explains things.
It is recommended on the ZR300's Intel online store listing though though that the Intel RealSense System Developer Kit (SDK) for Linux is used for the ZR300 camera, rather than the usual SDK used for the F200, R200 and SR300 cameras..
You can also use the open-source RealSense SDK, Librealsense.