For cuspidal automorphic π on GL_n (n≥3) and s with real part outside [1/n,1-1/n] (or [1/5,4/5] for n=4), L(s, π × χ) ≠ 0 for infinitely many primitive cubic Dirichlet characters χ.
On the periods of modular forms.Math
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Non-Vanishing of Cubic Twists of $GL_n(\mathbb{Q})$ $L$-functions
For cuspidal automorphic π on GL_n (n≥3) and s with real part outside [1/n,1-1/n] (or [1/5,4/5] for n=4), L(s, π × χ) ≠ 0 for infinitely many primitive cubic Dirichlet characters χ.