The early Z3 did not have an alarm as default from the factory. The early Z3 had an alarm fitted as a dealer option. My Z3 came with the EWS IIIG alarm. The IIIG alarm has a separate fob to control the alarm remotely. The alarm cannot be set by locking the car from the car doors. At some point in the past, the fob for the car was either lost or broken and no longer available to me. I can see that the alarm still seems to be active as it flashes when I stop the car and take out the key. It only flashes once and does not seem to be alarmed.
After months of searching around forums and eBay, someone posted a link to an eBay seller that had some IIIG fobs for sale. I quickly jumped on this and bought one.
I found a really useful post on zroadster.org that tells you how to sync the fob to the key
I hope that you find this information useful.