# Properties

 Label 2-18-3.2-c6-0-1 Degree $2$ Conductor $18$ Sign $-0.816 + 0.577i$ Analytic cond. $4.14097$ Root an. cond. $2.03493$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.65i·2-s − 32.0·4-s − 173. i·5-s − 484·7-s + 181. i·8-s − 984.·10-s − 1.34e3i·11-s + 3.36e3·13-s + 2.73e3i·14-s + 1.02e3·16-s − 12.7i·17-s + 5.74e3·19-s + 5.56e3i·20-s − 7.58e3·22-s + 3.37e3i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.500·4-s − 1.39i·5-s − 1.41·7-s + 0.353i·8-s − 0.984·10-s − 1.00i·11-s + 1.53·13-s + 0.997i·14-s + 0.250·16-s − 0.00259i·17-s + 0.837·19-s + 0.695i·20-s − 0.712·22-s + 0.277i·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}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, 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: $$4.14097$$ Root analytic conductor: $$2.03493$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{18} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 18,\ (\ :3),\ -0.816 + 0.577i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.329918 - 1.03801i$$ $$L(\frac12)$$ $$\approx$$ $$0.329918 - 1.03801i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 5.65iT$$
3 $$1$$
good5 $$1 + 173. iT - 1.56e4T^{2}$$
7 $$1 + 484T + 1.17e5T^{2}$$
11 $$1 + 1.34e3iT - 1.77e6T^{2}$$
13 $$1 - 3.36e3T + 4.82e6T^{2}$$
17 $$1 + 12.7iT - 2.41e7T^{2}$$
19 $$1 - 5.74e3T + 4.70e7T^{2}$$
23 $$1 - 3.37e3iT - 1.48e8T^{2}$$
29 $$1 + 2.93e4iT - 5.94e8T^{2}$$
31 $$1 + 3.97e4T + 8.87e8T^{2}$$
37 $$1 - 5.25e4T + 2.56e9T^{2}$$
41 $$1 - 3.70e4iT - 4.75e9T^{2}$$
43 $$1 - 3.80e3T + 6.32e9T^{2}$$
47 $$1 + 7.67e4iT - 1.07e10T^{2}$$
53 $$1 + 2.38e5iT - 2.21e10T^{2}$$
59 $$1 - 2.49e5iT - 4.21e10T^{2}$$
61 $$1 - 1.32e4T + 5.15e10T^{2}$$
67 $$1 - 1.68e5T + 9.04e10T^{2}$$
71 $$1 - 5.31e5iT - 1.28e11T^{2}$$
73 $$1 - 2.36e5T + 1.51e11T^{2}$$
79 $$1 + 3.51e4T + 2.43e11T^{2}$$
83 $$1 - 1.09e4iT - 3.26e11T^{2}$$
89 $$1 - 1.29e5iT - 4.96e11T^{2}$$
97 $$1 + 3.21e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$