An esoteric detective is recruited to find a pregnant woman’s killer, and a struggling rock and roll band holds the key to the mystery.

If you drop the recruiting section it gives you room for connective tissue. "Investigating a pregnant woman's murder a detective ....."

I personally have the feeling of a random sequence of unconnected elements... an esoteric detective, a murdered pregnant woman, a rnr band... This could be a?masterpiece (it makes me think about the novel Inherent Vice - I haven't seen the movie) but as a logline it fails to build interest because of a lack of coherence. The key of this one is 'the mystery': if the author really master his mystery he should try to give the reader some elements to prouve that.

As the others said.

Who recruits the detective? ?Who is the stakeholder, the one who must know who did it and bring the culprit to justice?

And if the detective is recruited, does that mean he's a private detective?