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You have a nice heirloom luger - keep it in the family !
You can confirm the caliber by trying to slide a standard #2 pencil into the barrel.
It will pass through a 9mm, but not the 7.65mm (.30 cal luger) barrel.
The ammunition is still in production.
Get it checked out by a gunsmith familiar with lugers.
I can't tell what kind of magazine it has, but if you shoot it, make sure it's a new magazine.
Mec-Gar still makes very good magazine and they are in many on-line gun stores.
The magazines work with 7.65mm or 9mm ammunition.
Use "target loads", not anything NATO or high powered.