C - E - Gb or C - E - F# ?

I thought first it was a Cdim, but I learned that the dim also changes the third to a minor, as aug changes the third to a major. How do you only change the fifth in a chord while keeping the minor or major of the chord?
Em6 formula is 1 - 3b - 5 - 6
I'm looking for how to name a chord with 1 - 3 - 5b
Thats an Em6 with a C as the bass note the chord your writing is not diatonicly related to C