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We prove the non-vanishing of special $L$-values of cuspidal automorphicforms on GL(2) twisted by Hecke characters of prime power orders and totallysplit prime power conductors. Main ingredients of the proof are estimating theGalois averages of fast convergent series expressions of special $L$-values andconsidering Shintani cone decomposition.

Non-vanishing of special $L$-values of cusp forms on GL(2) with totally split prime power twists