I had sex on the 9th of June, and on the 23rd of June. My due date that my doctor has given me is March 27th.
I am curious as to if its possible to narrow down my date of conception.

It's IMPOSSIBLE to know the exact date of conception as Gestational Dating is all based on Best Guess from available data.
If there is an issue surrounding paternity then a DNA test with all affected parties would be in order after the birth.

Thank you. I didn't think there was any way to narrow it down ,especially being so close together. I appreciate you taking the time to answer.

Those dates are much too close to give a good GUESS. If you said you only had sex once in June, then got a period, then had sex again in July, that'd be easier to venture a guess. ;)

Sperm can live inside your body for up to 7 days under the right conditions. That, and without tracking ovulation via basal body temps and charting means you can't say when you specifically ovulated. Eggs can live 24-ish hours. Implantation occurs several days later... the date your doctor gave is based on the first day of your last period and the size of your baby at the time of the scan along with the averages. The point is, a two week window means either day could have been conception day.

"Sperm can live inside your body for up to 7 days under the right conditions."

"Sperm ejaculated outside the body might survive in semen — the fluid released during ejaculation — up to a few hours."
Sperm: How long do they live after ejaculation? - MayoClinic.com

Sorry Learning-- I misread your post.

If ejaculated directly into fertile cervical fluid, (assuming all conditions are healthy), it can be viable for several days.

It's often intercourse can take place 2-3 days (or more) before the egg is released and "meets" sperm, resulting in conception.