On the non-vanishing of cubic twists of automorphic 𝐿-series

She, Xiaotie

doi:10.1090/S0002-9947-99-02082-6

On the non-vanishing of cubic twists of automorphic $L$-series
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by Xiaotie She

Trans. Amer. Math. Soc. 351 (1999), 1075-1094

DOI: https://doi.org/10.1090/S0002-9947-99-02082-6

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Abstract:

Let $f$ be a normalised new form of weight $2$ for $\Gamma _{0} (N)$ over ${\mathbb {Q}}$ and $F$, its base change lift to $\mathbb {Q}(\sqrt {-3})$. A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the $L$-function of $F$. There is an algorithm to check the condition for any given form. The new form of level $11$ is used to illustrate our method.

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