E50 single pilot reported deviation off their approach course at the FAF and receiving a low altitude warning from ATC.

Narrative

On Vectors to the Rnav RW XX at ZZZ. It was a long 11 hr day; with six legs; and I had not slept well the night before. I woke up at XA:00 local; and decided to prepare; instead of trying to get back to sleep to the alarm at XB:00. On the previous leg into ZZZ1; I had encountered some moderate- plus turbulence. That had given me a jolt of adrenaline that had not worn off by the time I briefly self-assessed before departing for the last leg of the day. On approach the sky was a thin layer of broken; and visibility was 3-10 nm in the area. It was going to be an easy approach at the end of a long day. I could see the airport as I flew past on vectors east; and let my guard down.ZZZ [TRACON] had vectored me just outside of the FAF. While I verbally briefed how I was going to set up for the approach miles back; I neglected to load vectors to final; so that G1000 was still thinking I would be heading outbound. As I was turned in outside of the FAF; cleared; and advised to maintain 3300 (I think) until established; the aircraft began a turn outbound instead of inbound. After the initial moment to recognize the issue; I turned off the autopilot; and the FD; and decided to hand fly the approach. I flew through course; and corrected. I had come out of the clouds and could see the airport; and the surrounding terrain; and began to neglect my scan. I thought I was closer to the FAF than I was; and so continued my descent. I got low; and ZZZ [TRACON] reached out to let me know I was low. I leveled off and climbed very gradually; as I had the airport in sight; and I did not want to affect passenger comfort. I neglected to tell ZZZ [TRACON] that I had the airport in sight. I flew level to capture Glidepath. It was a sloppy track all the way down. I hand fly approaches on empty legs to stay sharp. But as tired as I was- it was still a challenge. Contributing factors. Fatigue is number one. This was an eye opener for me. Complacency: Seeing the airport thinking I am there. Lack of communication- this comes back to fatigue- Aviate; Navigate; Communicate. I had just about enough bandwidth to do the first two; but not well. Get there Itis.Corrective actions. It is very rare for me to have this long of a day. In the future; I will have to be more careful about self-assessing on such a long day. I think a cup of coffee would have gone a long way to keep me sharp. I hadn't thought about how the adrenaline would wear off. I am truly not sure how to address the bad night of sleep. I usually sleep really well. There were a lot of legs; and I had prepped well for each one. Regardless of the weather - stick to procedures. This includes a constant review and practice of procedures; so that they are automatic.