Hi @Anastasia,
I'm afraid I do not understand what you mean by 'zero fine'. How was this information communicated to you by the Ausländerbehörde? How did they refer to it in German, by the word 'Geldstrafe' or 'Bußgeld'?
I can't see how having a fine would prevent your residence permit from being issued, unless it is money owed directly to the Ausländerbehörde, such as the fee for processing your application. Could this be the case? If not, are there any other municipal authorities you may owe money to, for example via the Stadtkasse? This is an office which collects money people owe for the public services in the area they live, for example charges for sending kids to a Kindergarten.
If the Ausländerbehörde are aware you have a fine to pay, then they will also be aware to which authoritiy the fine is to be paid. I advise you to request that information from the Ausländerbehörde.
Best,
Éanna