Properties

 Degree $2$ Conductor $1143$ Sign $0.577 + 0.816i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2.99i·2-s − 4.95·4-s + 7.40i·5-s − 4.26·7-s + 2.85i·8-s + 22.1·10-s − 11.1i·11-s − 7.91·13-s + 12.7i·14-s − 11.2·16-s − 14.3i·17-s + 18.0·19-s − 36.6i·20-s − 33.4·22-s + 33.5i·23-s + ⋯
 L(s)  = 1 − 1.49i·2-s − 1.23·4-s + 1.48i·5-s − 0.608·7-s + 0.356i·8-s + 2.21·10-s − 1.01i·11-s − 0.609·13-s + 0.910i·14-s − 0.704·16-s − 0.842i·17-s + 0.952·19-s − 1.83i·20-s − 1.52·22-s + 1.45i·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1143$$    =    $$3^{2} \cdot 127$$ Sign: $0.577 + 0.816i$ Motivic weight: $$2$$ Character: $\chi_{1143} (890, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1143,\ (\ :1),\ 0.577 + 0.816i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.499278544$$ $$L(\frac12)$$ $$\approx$$ $$1.499278544$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
127 $$1 - 11.2T$$
good2 $$1 + 2.99iT - 4T^{2}$$
5 $$1 - 7.40iT - 25T^{2}$$
7 $$1 + 4.26T + 49T^{2}$$
11 $$1 + 11.1iT - 121T^{2}$$
13 $$1 + 7.91T + 169T^{2}$$
17 $$1 + 14.3iT - 289T^{2}$$
19 $$1 - 18.0T + 361T^{2}$$
23 $$1 - 33.5iT - 529T^{2}$$
29 $$1 - 23.5iT - 841T^{2}$$
31 $$1 - 43.9T + 961T^{2}$$
37 $$1 - 2.75T + 1.36e3T^{2}$$
41 $$1 - 16.3iT - 1.68e3T^{2}$$
43 $$1 - 48.3T + 1.84e3T^{2}$$
47 $$1 - 4.06iT - 2.20e3T^{2}$$
53 $$1 - 74.2iT - 2.80e3T^{2}$$
59 $$1 + 55.4iT - 3.48e3T^{2}$$
61 $$1 - 45.3T + 3.72e3T^{2}$$
67 $$1 - 124.T + 4.48e3T^{2}$$
71 $$1 + 116. iT - 5.04e3T^{2}$$
73 $$1 - 93.3T + 5.32e3T^{2}$$
79 $$1 + 2.92T + 6.24e3T^{2}$$
83 $$1 + 72.3iT - 6.88e3T^{2}$$
89 $$1 - 19.0iT - 7.92e3T^{2}$$
97 $$1 - 18.2T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$