Properties

Label 2-162-9.2-c8-0-30
Degree $2$
Conductor $162$
Sign $-0.766 - 0.642i$
Analytic cond. $65.9953$
Root an. cond. $8.12375$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.79 − 5.65i)2-s + (63.9 + 110. i)4-s + (587. − 339. i)5-s + (1.03e3 − 1.78e3i)7-s − 1.44e3i·8-s − 7.68e3·10-s + (−5.76e3 − 3.32e3i)11-s + (−4.03e3 − 6.98e3i)13-s + (−2.02e4 + 1.16e4i)14-s + (−8.19e3 + 1.41e4i)16-s − 2.15e4i·17-s − 2.26e5·19-s + (7.52e4 + 4.34e4i)20-s + (3.76e4 + 6.51e4i)22-s + (−3.18e5 + 1.84e5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.940 − 0.543i)5-s + (0.430 − 0.744i)7-s − 0.353i·8-s − 0.768·10-s + (−0.393 − 0.227i)11-s + (−0.141 − 0.244i)13-s + (−0.526 + 0.304i)14-s + (−0.125 + 0.216i)16-s − 0.258i·17-s − 1.73·19-s + (0.470 + 0.271i)20-s + (0.160 + 0.278i)22-s + (−1.13 + 0.658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(65.9953\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :4),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.3062670709\)
\(L(\frac12)\) \(\approx\) \(0.3062670709\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.79 + 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (-587. + 339. i)T + (1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (-1.03e3 + 1.78e3i)T + (-2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (5.76e3 + 3.32e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + (4.03e3 + 6.98e3i)T + (-4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 + 2.15e4iT - 6.97e9T^{2} \)
19 \( 1 + 2.26e5T + 1.69e10T^{2} \)
23 \( 1 + (3.18e5 - 1.84e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (-8.11e5 - 4.68e5i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (4.13e5 + 7.15e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 - 1.34e6T + 3.51e12T^{2} \)
41 \( 1 + (4.49e6 - 2.59e6i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (-3.07e6 + 5.32e6i)T + (-5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (5.11e6 + 2.95e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + 7.68e5iT - 6.22e13T^{2} \)
59 \( 1 + (-4.10e5 + 2.36e5i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-7.49e6 + 1.29e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-5.01e6 - 8.68e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 4.54e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.32e7T + 8.06e14T^{2} \)
79 \( 1 + (7.13e6 - 1.23e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (3.13e7 + 1.80e7i)T + (1.12e15 + 1.95e15i)T^{2} \)
89 \( 1 - 1.15e8iT - 3.93e15T^{2} \)
97 \( 1 + (-2.02e7 + 3.51e7i)T + (-3.91e15 - 6.78e15i)T^{2} \)
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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

−10.50043984821670482468162120600, −9.885555799145481221998347050924, −8.723161172348619938743199877790, −7.87909665427348746062365363317, −6.57118734482855159547179272594, −5.30951718885098975359279722075, −4.01096097482061317383565361089, −2.35410875832564274099285127082, −1.34386033867077792913429051240, −0.084878001971269745288476880385, 1.80472632153134542664090237147, 2.52565491342958961405348416554, 4.59759683044826751584130098428, 5.94711591639743232410124412342, 6.57013352557210763496234707704, 8.029170318037459180900021840743, 8.835023029487585840671264500995, 10.03010979349929036365844764667, 10.59690251223476872411805995498, 11.84709822041518352100947909868

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