Properties

 Label 2-162-9.5-c6-0-4 Degree $2$ Conductor $162$ Sign $0.996 + 0.0871i$ Analytic cond. $37.2687$ Root an. cond. $6.10481$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−4.89 + 2.82i)2-s + (15.9 − 27.7i)4-s + (−180. − 104. i)5-s + (−2.09 − 3.63i)7-s + 181. i·8-s + 1.17e3·10-s + (−1.95e3 + 1.13e3i)11-s + (−1.42e3 + 2.45e3i)13-s + (20.5 + 11.8i)14-s + (−512. − 886. i)16-s + 1.96e3i·17-s − 281.·19-s + (−5.77e3 + 3.33e3i)20-s + (6.39e3 − 1.10e4i)22-s + (−1.45e4 − 8.37e3i)23-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.44 − 0.834i)5-s + (−0.00611 − 0.0105i)7-s + 0.353i·8-s + 1.17·10-s + (−1.47 + 0.849i)11-s + (−0.646 + 1.11i)13-s + (0.00749 + 0.00432i)14-s + (−0.125 − 0.216i)16-s + 0.400i·17-s − 0.0410·19-s + (−0.722 + 0.417i)20-s + (0.600 − 1.04i)22-s + (−1.19 − 0.688i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$162$$    =    $$2 \cdot 3^{4}$$ Sign: $0.996 + 0.0871i$ Analytic conductor: $$37.2687$$ Root analytic conductor: $$6.10481$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{162} (53, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 162,\ (\ :3),\ 0.996 + 0.0871i)$$

Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.4822505961$$ $$L(\frac12)$$ $$\approx$$ $$0.4822505961$$ $$L(4)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (4.89 - 2.82i)T$$
3 $$1$$
good5 $$1 + (180. + 104. i)T + (7.81e3 + 1.35e4i)T^{2}$$
7 $$1 + (2.09 + 3.63i)T + (-5.88e4 + 1.01e5i)T^{2}$$
11 $$1 + (1.95e3 - 1.13e3i)T + (8.85e5 - 1.53e6i)T^{2}$$
13 $$1 + (1.42e3 - 2.45e3i)T + (-2.41e6 - 4.18e6i)T^{2}$$
17 $$1 - 1.96e3iT - 2.41e7T^{2}$$
19 $$1 + 281.T + 4.70e7T^{2}$$
23 $$1 + (1.45e4 + 8.37e3i)T + (7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + (-3.21e4 + 1.85e4i)T + (2.97e8 - 5.15e8i)T^{2}$$
31 $$1 + (-1.23e4 + 2.13e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + 1.70e4T + 2.56e9T^{2}$$
41 $$1 + (1.00e5 + 5.81e4i)T + (2.37e9 + 4.11e9i)T^{2}$$
43 $$1 + (-1.53e4 - 2.65e4i)T + (-3.16e9 + 5.47e9i)T^{2}$$
47 $$1 + (6.72e4 - 3.88e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 - 1.38e5iT - 2.21e10T^{2}$$
59 $$1 + (-1.32e5 - 7.64e4i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (-8.06e3 - 1.39e4i)T + (-2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (-2.37e5 + 4.11e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + 1.50e5iT - 1.28e11T^{2}$$
73 $$1 - 3.31e5T + 1.51e11T^{2}$$
79 $$1 + (-4.48e5 - 7.76e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + (-8.18e5 + 4.72e5i)T + (1.63e11 - 2.83e11i)T^{2}$$
89 $$1 - 7.90e5iT - 4.96e11T^{2}$$
97 $$1 + (6.96e5 + 1.20e6i)T + (-4.16e11 + 7.21e11i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$