Per rules as written, the specter knows nothing of their previous life.
The relevant part of the description of thisthe Accursed Specter feature tells us that:
When you slay a humanoid, you can cause its spirit to rise from its corpse as a specter, the statistics for which are in the Monster Manual.
If one looks at the Specter entry in the MM, one can find (emphasis mine):
Specters no longer possess connections to who or what they were, yet are condemned to walk the world forever. Some are spawned when dark magic or the touch of a wraith rips a soul from a living body.
Since the description is part of the monster statistics, the bolded part says that actually the specter does not have any connection to their previous life, hence a strict reading states that he cannot answer to questions related to their crime activity.
As DM, you can overrule this.
Per rule 0, a DM can decide otherwise and let the specter be able to remember particulars from their previous life, and even to behave as they were in life:. I am referring to this enemy who:
talks back, goad the PCs to emotional, not caring if he's going to be killed or not, because at some point the organization will sent an evil cleric to revive him in the future, like in the past.
