You can download a copy of your owner's manual at the following link: http://www.chrysler.com/download/pdf/manuals/2012-200-OM-3rd.pdf
The owner's manual does not specify a requirement of conventional or synthetic type of oil:
For best performance and maximum protection under all
types of operating conditions, the manufacturer only
recommends engine oils that are API Certified and meet
the requirements of Chrysler Material Standard MS-6395.
MOPAR SAE 5W-20 engine oil or equivalent Pennzoil├é┬®
or Shell Helix├é┬® is recommended for all operating temperatures.
This engine oil improves low temperature
starting and vehicle fuel economy.
We recommend you use API Certified SAE 5W-20
Engine Oil, meeting the requirements of Chrysler Material
Standard MS-6395 such as MOPAR, Pennzoil├é┬®,
and Shell Helix├é┬®.
The engine oil filler cap also shows the recommended
engine oil viscosity for your engine.
You may use synthetic engine oils provided the recommended
oil quality requirements are met, and the recommended
maintenance intervals for oil and filter changes
Hope you find this helpful.