# Properties

 Label 2-162-3.2-c8-0-15 Degree $2$ Conductor $162$ Sign $-i$ Analytic cond. $65.9953$ Root an. cond. $8.12375$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 11.3i·2-s − 128.·4-s + 618. i·5-s − 549.·7-s − 1.44e3i·8-s − 6.99e3·10-s − 1.92e4i·11-s + 2.08e3·13-s − 6.21e3i·14-s + 1.63e4·16-s + 5.81e4i·17-s + 2.02e5·19-s − 7.91e4i·20-s + 2.18e5·22-s − 6.90e4i·23-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.500·4-s + 0.988i·5-s − 0.228·7-s − 0.353i·8-s − 0.699·10-s − 1.31i·11-s + 0.0729·13-s − 0.161i·14-s + 0.250·16-s + 0.696i·17-s + 1.55·19-s − 0.494i·20-s + 0.930·22-s − 0.246i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$162$$    =    $$2 \cdot 3^{4}$$ Sign: $-i$ Analytic conductor: $$65.9953$$ Root analytic conductor: $$8.12375$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{162} (161, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 162,\ (\ :4),\ -i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.897046487$$ $$L(\frac12)$$ $$\approx$$ $$1.897046487$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 11.3iT$$
3 $$1$$
good5 $$1 - 618. iT - 3.90e5T^{2}$$
7 $$1 + 549.T + 5.76e6T^{2}$$
11 $$1 + 1.92e4iT - 2.14e8T^{2}$$
13 $$1 - 2.08e3T + 8.15e8T^{2}$$
17 $$1 - 5.81e4iT - 6.97e9T^{2}$$
19 $$1 - 2.02e5T + 1.69e10T^{2}$$
23 $$1 + 6.90e4iT - 7.83e10T^{2}$$
29 $$1 + 3.62e5iT - 5.00e11T^{2}$$
31 $$1 - 5.45e4T + 8.52e11T^{2}$$
37 $$1 - 6.32e5T + 3.51e12T^{2}$$
41 $$1 + 5.05e6iT - 7.98e12T^{2}$$
43 $$1 - 1.99e6T + 1.16e13T^{2}$$
47 $$1 - 7.04e6iT - 2.38e13T^{2}$$
53 $$1 - 4.75e6iT - 6.22e13T^{2}$$
59 $$1 - 6.52e6iT - 1.46e14T^{2}$$
61 $$1 - 9.32e6T + 1.91e14T^{2}$$
67 $$1 - 1.17e7T + 4.06e14T^{2}$$
71 $$1 - 3.81e7iT - 6.45e14T^{2}$$
73 $$1 - 5.36e7T + 8.06e14T^{2}$$
79 $$1 + 5.15e7T + 1.51e15T^{2}$$
83 $$1 - 3.30e7iT - 2.25e15T^{2}$$
89 $$1 + 1.36e7iT - 3.93e15T^{2}$$
97 $$1 + 1.10e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$