# Properties

 Label 2-1134-63.4-c1-0-28 Degree $2$ Conductor $1134$ Sign $-0.0788 + 0.996i$ Analytic cond. $9.05503$ Root an. cond. $3.00915$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 3·5-s + (−2 − 1.73i)7-s − 0.999·8-s + (1.5 − 2.59i)10-s + 3·11-s + (2 − 3.46i)13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)19-s + (−1.49 − 2.59i)20-s + (1.5 − 2.59i)22-s + 4·25-s + (−1.99 − 3.46i)26-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.34·5-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (0.474 − 0.821i)10-s + 0.904·11-s + (0.554 − 0.960i)13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.458 + 0.794i)19-s + (−0.335 − 0.580i)20-s + (0.319 − 0.553i)22-s + 0.800·25-s + (−0.392 − 0.679i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1134$$    =    $$2 \cdot 3^{4} \cdot 7$$ Sign: $-0.0788 + 0.996i$ Analytic conductor: $$9.05503$$ Root analytic conductor: $$3.00915$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1134} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1134,\ (\ :1/2),\ -0.0788 + 0.996i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.310142906$$ $$L(\frac12)$$ $$\approx$$ $$2.310142906$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
3 $$1$$
7 $$1 + (2 + 1.73i)T$$
good5 $$1 - 3T + 5T^{2}$$
11 $$1 - 3T + 11T^{2}$$
13 $$1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$