Properties

Label 2-162-9.2-c6-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−132. + 76.7i)5-s + (−336. + 582. i)7-s − 181. i·8-s + 868.·10-s + (2.21e3 + 1.27e3i)11-s + (885. + 1.53e3i)13-s + (3.29e3 − 1.90e3i)14-s + (−512. + 886. i)16-s + 3.97e3i·17-s − 7.76e3·19-s + (−4.25e3 − 2.45e3i)20-s + (−7.24e3 − 1.25e4i)22-s + (2.26e3 − 1.30e3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.06 + 0.614i)5-s + (−0.979 + 1.69i)7-s − 0.353i·8-s + 0.868·10-s + (1.66 + 0.961i)11-s + (0.402 + 0.697i)13-s + (1.20 − 0.692i)14-s + (−0.125 + 0.216i)16-s + 0.809i·17-s − 1.13·19-s + (−0.531 − 0.307i)20-s + (−0.679 − 1.17i)22-s + (0.186 − 0.107i)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: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ -0.996 + 0.0871i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7733886340\)
\(L(\frac12)\) \(\approx\) \(0.7733886340\)
\(L(4)\) 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 \( 1 + (132. - 76.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (336. - 582. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-2.21e3 - 1.27e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-885. - 1.53e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 3.97e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.76e3T + 4.70e7T^{2} \)
23 \( 1 + (-2.26e3 + 1.30e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.25e4 - 7.23e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-2.22e4 - 3.84e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + 3.95e4T + 2.56e9T^{2} \)
41 \( 1 + (1.41e4 - 8.15e3i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (2.76e4 - 4.78e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.03e5 - 5.96e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 5.93e3iT - 2.21e10T^{2} \)
59 \( 1 + (-6.43e3 + 3.71e3i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (2.72e4 - 4.72e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.11e5 - 1.93e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 7.24e4iT - 1.28e11T^{2} \)
73 \( 1 + 3.17e5T + 1.51e11T^{2} \)
79 \( 1 + (8.64e4 - 1.49e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (1.26e4 + 7.33e3i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + 8.00e5iT - 4.96e11T^{2} \)
97 \( 1 + (-8.39e4 + 1.45e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−12.11178074301917587591024502869, −11.45218202011484169777886014117, −10.19329418386973567251185907495, −9.049029143138030244730118012747, −8.544535601729318052999936709369, −6.89543601190936714985204049269, −6.32978437469452089316587136933, −4.20053769388336073935848969369, −3.11481225569883797997364361747, −1.75422484896071429779228433891, 0.39541696214694971454449522988, 0.858829791280845570190690621125, 3.51727341195596417855306126526, 4.29342394411152670400902565185, 6.23264429642926248802631093230, 7.07107814908178845288083877149, 8.133593610475344373053087741321, 9.041177520575314434163942839872, 10.16904218364551561799428375233, 11.12601017949002621685482598792

Graph of the $Z$-function along the critical line