# Properties

 Label 2-1183-91.18-c0-0-0 Degree $2$ Conductor $1183$ Sign $0.580 - 0.814i$ Analytic cond. $0.590393$ Root an. cond. $0.768370$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.965 − 0.258i)2-s + (−0.5 + 0.866i)3-s + (0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)10-s + (0.965 − 0.258i)11-s + (−0.500 − 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.965 + 0.258i)19-s + (−0.965 + 0.258i)21-s − 22-s + (0.866 − 0.5i)23-s + ⋯
 L(s)  = 1 + (−0.965 − 0.258i)2-s + (−0.5 + 0.866i)3-s + (0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)10-s + (0.965 − 0.258i)11-s + (−0.500 − 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.965 + 0.258i)19-s + (−0.965 + 0.258i)21-s − 22-s + (0.866 − 0.5i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1183$$    =    $$7 \cdot 13^{2}$$ Sign: $0.580 - 0.814i$ Analytic conductor: $$0.590393$$ Root analytic conductor: $$0.768370$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1183} (746, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1183,\ (\ :0),\ 0.580 - 0.814i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6958128246$$ $$L(\frac12)$$ $$\approx$$ $$0.6958128246$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-0.707 - 0.707i)T$$
13 $$1$$
good2 $$1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}$$
3 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
5 $$1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}$$
11 $$1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}$$
17 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
19 $$1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}$$
23 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}$$
37 $$1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}$$
41 $$1 - iT^{2}$$
43 $$1 - T^{2}$$
47 $$1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}$$
61 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2}$$
71 $$1 - iT^{2}$$
73 $$1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}$$
79 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}$$
97 $$1 + (1.41 - 1.41i)T - iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$