Luke uniquely said that these Jewish elders continued praising this centurion, who loved the Jewish people, their people (ἀγαπᾷ γὰρ τὸ ἔθνος ἡμῶν). He had built a synagogue for them (καὶ τὴν συναγωγὴν αὐτὸς ᾠκοδόμησεν ἡμῖν). There were many instances of Roman soldiers adopting the religious practices of the people where they were staying. However, building a synagogue seems a bit much. It may have led to better community relations. Although he was not Jewish, this centurion had been very favorable to the Jewish people by helping them build a new synagogue. There was no mention of this synagogue in the Matthew story about the centurion. Would you be favorable to a religion not your own?
The Zadok Levitical priests had strict dietary habits that were very favorable to them. They were to eat all the various offerings that the people of Israel presented, the grain offerings, the sin offerings, the guilt offerings, or any devoted thing in Israel. They would also get the first fruits of all kinds, including the cooked dough. All these food items belonged to the priests. Giving them to the priests meant that your house would have a blessing. However, these priests were not allowed to eat any bird or animal that had died by itself or was mangled by another animal.