That C (Τὸ Ϲίγμα τοῦτο) is called a lunate sigma (because it looks like a crescent). Per Wikipedia, https://en.wikipedia.org/wiki/Si…:
In handwritten Greek during the Hellenistic period (4th and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape. It is also found on coins from the fourth century BC onward. This became the universal standard form of sigma during late antiquity and the Middle Ages.
I wrote a little more on the lunate sigma in my site on Unicode Greek:
While in the 4th century BC literary papyri still used the ancient angular sigma (Thompson 1912:107), the cursive lunate form had taken over completely by the next century, where it stayed as the unique form of the sigma until the invention of lowercase in the 8th century—and as the capital sigma, for a millenium longer. […]
While the lunate was banished as a capital letter in the 18th century, it remains familiar to Modern Greeks through ecclesiastical use: it figures in church icons, and in decorative fonts intended to evoke Byzantium.
Btw, no reason to think it was under Roman influence. Not that early.