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Non-vanishing of derivatives of L-functions associated to cusp forms of half-integral weight in the plus space

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Abstract

We show a non-vanishing result for the derivatives of L-functions associated to cuspidal Hecke eigenforms of half-integral weight in plus space. In particular, we show that for large weights,

$$\begin{aligned} \sum _{j=1}^{d}\frac{1}{\langle f_{k,j}, f_{k,j} \rangle }\frac{d^n}{ds^n}[L^*(f_{k,j}|W_4,s)] \end{aligned}$$

does not vanish at any point \(s=\sigma +it_0\) with \(t=t_0,k/2-1/4<\sigma <k/2+3/4\).

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Acknowledgements

I would like to thank Professor Winfried Kohnen for his valuable comments. In particular, Professor Kohnen pointed out that in [4], the result follows for k odd as well by considering the third Fourier coefficient instead of the first Fourier coefficient.

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  1. American University of Beirut, Beirut, Lebanon

    Wissam Raji

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  1. Wissam Raji
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Correspondence to Wissam Raji.

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Raji, W. Non-vanishing of derivatives of L-functions associated to cusp forms of half-integral weight in the plus space. Ramanujan J 62, 533–543 (2023). https://doi.org/10.1007/s11139-022-00641-w

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  • DOI: https://doi.org/10.1007/s11139-022-00641-w

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