Properties

 Label 2-33e2-3.2-c2-0-51 Degree $2$ Conductor $1089$ Sign $-0.577 + 0.816i$ Analytic cond. $29.6731$ Root an. cond. $5.44730$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2.03i·2-s − 0.127·4-s + 0.796i·5-s + 1.12·7-s − 7.86i·8-s + 1.61·10-s + 9.25·13-s − 2.28i·14-s − 16.4·16-s − 11.7i·17-s + 17.2·19-s − 0.101i·20-s − 5.11i·23-s + 24.3·25-s − 18.7i·26-s + ⋯
 L(s)  = 1 − 1.01i·2-s − 0.0317·4-s + 0.159i·5-s + 0.161·7-s − 0.983i·8-s + 0.161·10-s + 0.711·13-s − 0.163i·14-s − 1.03·16-s − 0.691i·17-s + 0.907·19-s − 0.00506i·20-s − 0.222i·23-s + 0.974·25-s − 0.723i·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1089$$    =    $$3^{2} \cdot 11^{2}$$ Sign: $-0.577 + 0.816i$ Analytic conductor: $$29.6731$$ Root analytic conductor: $$5.44730$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1089} (485, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1089,\ (\ :1),\ -0.577 + 0.816i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.218807024$$ $$L(\frac12)$$ $$\approx$$ $$2.218807024$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$1 + 2.03iT - 4T^{2}$$
5 $$1 - 0.796iT - 25T^{2}$$
7 $$1 - 1.12T + 49T^{2}$$
13 $$1 - 9.25T + 169T^{2}$$
17 $$1 + 11.7iT - 289T^{2}$$
19 $$1 - 17.2T + 361T^{2}$$
23 $$1 + 5.11iT - 529T^{2}$$
29 $$1 - 29.0iT - 841T^{2}$$
31 $$1 + 52.7T + 961T^{2}$$
37 $$1 - 37.3T + 1.36e3T^{2}$$
41 $$1 + 75.0iT - 1.68e3T^{2}$$
43 $$1 - 68.8T + 1.84e3T^{2}$$
47 $$1 + 9.36iT - 2.20e3T^{2}$$
53 $$1 - 34.0iT - 2.80e3T^{2}$$
59 $$1 + 69.2iT - 3.48e3T^{2}$$
61 $$1 + 24.9T + 3.72e3T^{2}$$
67 $$1 + 16.8T + 4.48e3T^{2}$$
71 $$1 + 77.0iT - 5.04e3T^{2}$$
73 $$1 - 55.2T + 5.32e3T^{2}$$
79 $$1 - 64.9T + 6.24e3T^{2}$$
83 $$1 + 120. iT - 6.88e3T^{2}$$
89 $$1 + 110. iT - 7.92e3T^{2}$$
97 $$1 + 14.0T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$