I don't have a product serial number

The serial number can be found inside one of the channels where your watch band goes in. It can often be found on the back of the case and it can be found using the Watch App on your iPhone, which the watch is paired with.

See this --> How to find the serial number or IMEI for your Apple Watch - Apple Support

OK.....did you open the link and see where in the watch band track the serial number can be found? I also said you can look there in my first reply. You may need a magnifying glass to see it, but it is there.