How to generate RSA with SHA512 using a key?

Question

Hashing

I have an RSA public and private key from a service certificate.

I need to encode my data with rsa using sha512 using the public key and compare it with the received parameter checksum.
If the encoded data is equal to the checksum, the operation was successful.
I looked for an online coding tool to check all this, but

1. I can't use any hash key
2. The size of the cheksum is many times larger than the rsa sha512 encoded data, for example:

checksum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

data='mdOrder;2230808c-6fa8-71f5-b6ff-54085e4149db;operation;deposited;orderNumber;#19;sign_alias;stoloto;status;1;
encoded_data='CA0D0D2B7EFC86C56CE098A5F3D684C850F3918A9E7F1DB75397A06360CB200C'

I suppose I do not have enough knowledge about cryptography and I ask you to explain to me how to do this, the service wrote this: