And if you want to work out what your take-home pay will be then the Öffentlicher-Dienst.info
site has calculators, amongst much other info on the tarifs for the German civil service.
Incidentally, the /2 may not (in fact, probably does not) mean Stufe 2, but a half-time post (i.e. TV-L E13 divided by 2) which will of course only be paid half the full salary. It is usual for "Doktoranden" in Germany to be paid as a part-time (usually between 50 and 75%, i.e. 20-30 hrs/wk) post, and research projects and PhDs are funded by e.g. BMBF and DFG on that basis. Whether you will only be expected to work part-time or actually full-time is between you and the Prof and/or the other colleagues (often you may be expected to spend the rest of the time on your actual PhD rather than doing the project donkey work).
See also this earlier thread
on the subject - from post 18
then pretty much this exact topic was discussed.