The investigation the detector is going to make/do to find out who committed the crime.
Which is more idiomatic, to make or to do an investigation?
Investigations are usually conducted. Conduct an investigation is what we call a collocation in English. A collocation is basically a naturally occurring combination of words that are always used side by side in a particular context which means that collocations are by default idiomatic.
An alternative to the expression conduct an investigation that will probably work even better in your example would be to launch an investigation:
The investigation the detective is going to launch to find out who committed the crime is going to take about eight months.
I think that Investigate alone will be the best choice for that specific sentence. I would write:
The detective will investigate to find out who committed the crime.
Investigate is already a verb which means "carry out a systematic or formal inquiry to discover and examine the facts of (an incident, allegation, etc.) so as to establish the truth."
You CAN use conduct or launch to form the sentence:
The detective will launch/conduct an investigation to find out who committed the crime.
and that would be best if you are talking specifically about the action of STARTING the investigation. If you are talking about the investigation itself, I would leave out conduct/launch/make/do/etc, because they make the sentence a bit wordy.
Lastly, as a bit of an aside, I would shy away from that "double the". Your sentence starts with
The investigation the [detective] is going to launch/conduct...
which sounds off. You can keep the same format by adding in "that" between the "the" clauses, making it:
The investigation that the [detective] is going to launch/conduct...