Non-vanishing theorems for prime twists of some modular $L$-functions

In this paper, we give some non-vanishing results on the central values of prime twists of modular $L$-functions by imaginary quadratic fields for specific elliptic modular forms. In particular, we show that the central values of prime twists of $L$-functions of some elliptic modular forms are always non-vanishing whenever the root number is positive.