# Properties

 Label 2-162-3.2-c8-0-14 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 + 1.06e3i·5-s − 3.23e3·7-s + 1.44e3i·8-s + 1.20e4·10-s − 7.89e3i·11-s − 5.44e4·13-s + 3.65e4i·14-s + 1.63e4·16-s + 8.53e4i·17-s − 1.53e5·19-s − 1.35e5i·20-s − 8.92e4·22-s + 3.52e5i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.500·4-s + 1.69i·5-s − 1.34·7-s + 0.353i·8-s + 1.20·10-s − 0.539i·11-s − 1.90·13-s + 0.952i·14-s + 0.250·16-s + 1.02i·17-s − 1.18·19-s − 0.849i·20-s − 0.381·22-s + 1.25i·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

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.3902221555$$ $$L(\frac12)$$ $$\approx$$ $$0.3902221555$$ $$L(5)$$ 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 - 1.06e3iT - 3.90e5T^{2}$$
7 $$1 + 3.23e3T + 5.76e6T^{2}$$
11 $$1 + 7.89e3iT - 2.14e8T^{2}$$
13 $$1 + 5.44e4T + 8.15e8T^{2}$$
17 $$1 - 8.53e4iT - 6.97e9T^{2}$$
19 $$1 + 1.53e5T + 1.69e10T^{2}$$
23 $$1 - 3.52e5iT - 7.83e10T^{2}$$
29 $$1 + 8.80e5iT - 5.00e11T^{2}$$
31 $$1 - 1.43e6T + 8.52e11T^{2}$$
37 $$1 - 1.22e6T + 3.51e12T^{2}$$
41 $$1 + 2.91e5iT - 7.98e12T^{2}$$
43 $$1 - 2.30e6T + 1.16e13T^{2}$$
47 $$1 - 1.84e6iT - 2.38e13T^{2}$$
53 $$1 + 3.72e6iT - 6.22e13T^{2}$$
59 $$1 + 3.50e6iT - 1.46e14T^{2}$$
61 $$1 + 4.73e4T + 1.91e14T^{2}$$
67 $$1 + 8.18e6T + 4.06e14T^{2}$$
71 $$1 - 6.23e6iT - 6.45e14T^{2}$$
73 $$1 - 3.16e7T + 8.06e14T^{2}$$
79 $$1 + 2.77e7T + 1.51e15T^{2}$$
83 $$1 - 8.77e7iT - 2.25e15T^{2}$$
89 $$1 + 8.43e7iT - 3.93e15T^{2}$$
97 $$1 - 2.33e7T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$