# Properties

 Label 2-1134-21.20-c1-0-19 Degree $2$ Conductor $1134$ Sign $0.861 - 0.506i$ 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 + i·2-s − 4-s + 1.79·5-s + (2.28 − 1.34i)7-s − i·8-s + 1.79i·10-s − 2.40i·11-s + 4.89i·13-s + (1.34 + 2.28i)14-s + 16-s + 3.66·17-s − 3.01i·19-s − 1.79·20-s + 2.40·22-s − 3.76i·23-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 0.800·5-s + (0.861 − 0.506i)7-s − 0.353i·8-s + 0.566i·10-s − 0.724i·11-s + 1.35i·13-s + (0.358 + 0.609i)14-s + 0.250·16-s + 0.888·17-s − 0.692i·19-s − 0.400·20-s + 0.512·22-s − 0.785i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1134$$    =    $$2 \cdot 3^{4} \cdot 7$$ Sign: $0.861 - 0.506i$ Analytic conductor: $$9.05503$$ Root analytic conductor: $$3.00915$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1134} (1133, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1134,\ (\ :1/2),\ 0.861 - 0.506i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.006151952$$ $$L(\frac12)$$ $$\approx$$ $$2.006151952$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1$$
7 $$1 + (-2.28 + 1.34i)T$$
good5 $$1 - 1.79T + 5T^{2}$$
11 $$1 + 2.40iT - 11T^{2}$$
13 $$1 - 4.89iT - 13T^{2}$$
17 $$1 - 3.66T + 17T^{2}$$
19 $$1 + 3.01iT - 19T^{2}$$
23 $$1 + 3.76iT - 23T^{2}$$
29 $$1 + 6.56iT - 29T^{2}$$
31 $$1 - 4.64iT - 31T^{2}$$
37 $$1 - 9.36T + 37T^{2}$$
41 $$1 - 8.08T + 41T^{2}$$
43 $$1 - 6.96T + 43T^{2}$$
47 $$1 + 5.13T + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 14.5T + 59T^{2}$$
61 $$1 - 11.3iT - 61T^{2}$$
67 $$1 - 0.570T + 67T^{2}$$
71 $$1 + 5.96iT - 71T^{2}$$
73 $$1 - 12.3iT - 73T^{2}$$
79 $$1 - 3.03T + 79T^{2}$$
83 $$1 - 14.0T + 83T^{2}$$
89 $$1 + 3.74T + 89T^{2}$$
97 $$1 - 5.51iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$