# Properties

 Label 2-18-3.2-c2-0-1 Degree $2$ Conductor $18$ Sign $0.816 + 0.577i$ Analytic cond. $0.490464$ Root an. cond. $0.700331$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s − 2.00·4-s + 4.24i·5-s − 4·7-s + 2.82i·8-s + 6·10-s − 16.9i·11-s + 8·13-s + 5.65i·14-s + 4.00·16-s + 12.7i·17-s − 16·19-s − 8.48i·20-s − 24·22-s + 16.9i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.500·4-s + 0.848i·5-s − 0.571·7-s + 0.353i·8-s + 0.600·10-s − 1.54i·11-s + 0.615·13-s + 0.404i·14-s + 0.250·16-s + 0.748i·17-s − 0.842·19-s − 0.424i·20-s − 1.09·22-s + 0.737i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$18$$    =    $$2 \cdot 3^{2}$$ Sign: $0.816 + 0.577i$ Analytic conductor: $$0.490464$$ Root analytic conductor: $$0.700331$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{18} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 18,\ (\ :1),\ 0.816 + 0.577i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.751341 - 0.238804i$$ $$L(\frac12)$$ $$\approx$$ $$0.751341 - 0.238804i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41iT$$
3 $$1$$
good5 $$1 - 4.24iT - 25T^{2}$$
7 $$1 + 4T + 49T^{2}$$
11 $$1 + 16.9iT - 121T^{2}$$
13 $$1 - 8T + 169T^{2}$$
17 $$1 - 12.7iT - 289T^{2}$$
19 $$1 + 16T + 361T^{2}$$
23 $$1 - 16.9iT - 529T^{2}$$
29 $$1 + 4.24iT - 841T^{2}$$
31 $$1 - 44T + 961T^{2}$$
37 $$1 + 34T + 1.36e3T^{2}$$
41 $$1 + 46.6iT - 1.68e3T^{2}$$
43 $$1 + 40T + 1.84e3T^{2}$$
47 $$1 - 84.8iT - 2.20e3T^{2}$$
53 $$1 + 38.1iT - 2.80e3T^{2}$$
59 $$1 + 33.9iT - 3.48e3T^{2}$$
61 $$1 - 50T + 3.72e3T^{2}$$
67 $$1 - 8T + 4.48e3T^{2}$$
71 $$1 - 50.9iT - 5.04e3T^{2}$$
73 $$1 + 16T + 5.32e3T^{2}$$
79 $$1 + 76T + 6.24e3T^{2}$$
83 $$1 + 118. iT - 6.88e3T^{2}$$
89 $$1 + 12.7iT - 7.92e3T^{2}$$
97 $$1 - 176T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$