The verb ἐλεγχω occurs as an adjectival participle (nom masc sing pres pass) describing Herod’s response to John the Baptist’s preaching ministry (cf. 3:3). Looking up this verb in BDAG reveals something interesting.
BDAG lists this occurrence of this verb under meaning 3 of ἐλεγχω which is “to express stong disapproval of someone’s action.” This might sound like a very good definition of this word – since we go on to read that Herod threw John in prison.
However, I think that BDAG might have mis-categorised this occurrence. Meaning 2 given by BDAG for this verb is “to bring a pers. to the point of recognizing wrongdoing” – a suitable gloss might be ‘to bring conviction’. I think this definition suits this occurrence much better for two reasons:
Firstly because of the wider context where John is preaching/teaching about the need for repentance. So it makes sense that John’s preaching brings conviction with it.
Secondly, however, is the immediate clause structure. Under meaning 2, BDAG lists περι + gen as being commonly used to introduce “the thing of which one is convicted or convinced”. That is the exact structure that we have in Luke 3:19. The participle is followed by two parallel καί-conjoined prepositional phrases containing περι + gen which do exactly what BDAG suggests – they describe what it is that Herod feels conviction about.