The interaction could be very comical. Picture Conan being confused by AI Lemon's overly technical explanations. AI Lemon might say something like 'The probability matrix indicates a 78.5% chance of the clue being in this location based on these 12 variables' and Conan just stares blankly for a moment before asking for a simpler explanation. This would add a light - hearted touch to their relationship in the fanfiction.
Their interaction can be portrayed as a partnership. Conan could talk to AI Lemon like he would with any of his human friends, asking for advice or information. For example, he might say 'AI Lemon, what do you know about this suspect's digital footprint?' and AI Lemon would respond with relevant data.