A note on simultaneous nonvanishing twists

Given two Hecke cusp forms f1 and f2 of SL(2,Z). Suppose there is a quadratic character χ such that the twisted L-functions L(s,fi⊗χ) do not vanish at the center s=1/2. Then we show that there are infinitely many primitive quadratic characters χd such that L(1/2,f1⊗χd)L(1/2,f2⊗χd)≠0.