## Coefficients of Maass forms and the Siegel zero. Appendix: An effective zero-free region, by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman.(English)Zbl 0814.11032

Let $$f$$ be a Maaß form, which is a newform for $$\Gamma_ 0(N)$$ with eigenvalue $$\lambda$$ and Dirichlet character $$\chi \bmod N$$, normalized such that the Petersson inner product is 1. Let $$F$$ denote the adjoint square lift of $$f$$ to $$\text{GL}(3)$$. It is known that the size of the first Fourier coefficients $$\rho(1)$$ of $$f$$ is closely related to the behavior of the $$L$$-series $$L(s,F)$$ near $$s = 1$$.
In Theorem 1 the authors show that $| \rho(1)|^ 2 \leq c \log (\lambda N + 1)$ with an effective constant $$c$$ provided that no Siegel zero occurs, i.e. $$L(s,F) \neq 0$$ in a sufficiently large neighborhood of 1. In the appendix the same estimate is derived for all $$f$$ which are not lifts from $$\text{GL}(1)$$, i.e. the $$L$$-series of $$f$$ is not equal to a Hecke $$L$$-series of a quadratic field. A weaker estimate for lifts from $$\text{GL}(1)$$ is also included.
The paper goes on with the inequality $$L(1,F) \geq c(\varepsilon) (\lambda N)^{-\varepsilon}$$ for all $$\varepsilon > 0$$ with an effective constant $$c(\varepsilon)$$ and for all $$F$$ with one possible exception. This in turn leads to $$| \rho(1)| \ll_ \varepsilon (\lambda N)^ \varepsilon$$.
Reviewer: A.Krieg (Aachen)

### MSC:

 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F37 Forms of half-integer weight; nonholomorphic modular forms
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