# Properties

 Label 2-162-3.2-c8-0-27 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 $0$

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## Dirichlet series

 L(s)  = 1 + 11.3i·2-s − 128.·4-s − 654. i·5-s + 1.29e3·7-s − 1.44e3i·8-s + 7.40e3·10-s − 4.40e3i·11-s − 246.·13-s + 1.47e4i·14-s + 1.63e4·16-s − 8.90e4i·17-s + 1.45e5·19-s + 8.37e4i·20-s + 4.98e4·22-s + 3.55e5i·23-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.500·4-s − 1.04i·5-s + 0.541·7-s − 0.353i·8-s + 0.740·10-s − 0.301i·11-s − 0.00861·13-s + 0.382i·14-s + 0.250·16-s − 1.06i·17-s + 1.11·19-s + 0.523i·20-s + 0.212·22-s + 1.27i·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

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.394754426$$ $$L(\frac12)$$ $$\approx$$ $$1.394754426$$ $$L(5)$$ 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 + 654. iT - 3.90e5T^{2}$$
7 $$1 - 1.29e3T + 5.76e6T^{2}$$
11 $$1 + 4.40e3iT - 2.14e8T^{2}$$
13 $$1 + 246.T + 8.15e8T^{2}$$
17 $$1 + 8.90e4iT - 6.97e9T^{2}$$
19 $$1 - 1.45e5T + 1.69e10T^{2}$$
23 $$1 - 3.55e5iT - 7.83e10T^{2}$$
29 $$1 - 6.35e4iT - 5.00e11T^{2}$$
31 $$1 + 9.72e5T + 8.52e11T^{2}$$
37 $$1 + 4.08e5T + 3.51e12T^{2}$$
41 $$1 + 3.59e6iT - 7.98e12T^{2}$$
43 $$1 + 4.83e6T + 1.16e13T^{2}$$
47 $$1 - 1.25e6iT - 2.38e13T^{2}$$
53 $$1 + 1.23e7iT - 6.22e13T^{2}$$
59 $$1 + 3.30e6iT - 1.46e14T^{2}$$
61 $$1 - 1.51e7T + 1.91e14T^{2}$$
67 $$1 + 1.76e7T + 4.06e14T^{2}$$
71 $$1 + 2.66e7iT - 6.45e14T^{2}$$
73 $$1 + 1.28e7T + 8.06e14T^{2}$$
79 $$1 - 5.49e7T + 1.51e15T^{2}$$
83 $$1 + 9.14e6iT - 2.25e15T^{2}$$
89 $$1 - 5.99e7iT - 3.93e15T^{2}$$
97 $$1 - 1.07e8T + 7.83e15T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−11.26429441904918535027431173198, −9.718698351105912329435091069472, −8.955349121603095698394274268134, −7.977987845344614154018112228320, −7.04006748655955136255151245436, −5.44408177573993765723074110979, −4.96622159822731135247967564470, −3.49317481692258953464687564948, −1.53092308033709128563808825005, −0.35595863417613802871883126837, 1.35074560142666889127378069781, 2.55611855536053374170899808270, 3.65585765890095961972491350134, 4.94418041718299160267072052098, 6.33811071964865885038713283558, 7.51920531296853240942023673201, 8.614552636997785830898385132934, 9.904885100501877627620946674657, 10.67579225061971742916399660718, 11.44199592044889091871411513110