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Properties

Label 2-180-4.3-c8-0-57
Degree $2$
Conductor $180$
Sign $0.371 + 0.928i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
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  + (−8.96 + 13.2i)2-s + (−95.1 − 237. i)4-s − 279.·5-s − 3.40e3i·7-s + (4.00e3 + 869. i)8-s + (2.50e3 − 3.70e3i)10-s − 3.24e3i·11-s + 1.00e4·13-s + (4.50e4 + 3.04e4i)14-s + (−4.74e4 + 4.52e4i)16-s + 1.12e5·17-s − 6.21e4i·19-s + (2.66e4 + 6.64e4i)20-s + (4.30e4 + 2.91e4i)22-s + 2.50e5i·23-s + ⋯
L(s)  = 1  + (−0.560 + 0.828i)2-s + (−0.371 − 0.928i)4-s − 0.447·5-s − 1.41i·7-s + (0.977 + 0.212i)8-s + (0.250 − 0.370i)10-s − 0.221i·11-s + 0.350·13-s + (1.17 + 0.793i)14-s + (−0.723 + 0.690i)16-s + 1.34·17-s − 0.477i·19-s + (0.166 + 0.415i)20-s + (0.183 + 0.124i)22-s + 0.894i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.159817251\)
\(L(\frac12)\) \(\approx\) \(1.159817251\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8.96 - 13.2i)T \)
3 \( 1 \)
5 \( 1 + 279.T \)
good7 \( 1 + 3.40e3iT - 5.76e6T^{2} \)
11 \( 1 + 3.24e3iT - 2.14e8T^{2} \)
13 \( 1 - 1.00e4T + 8.15e8T^{2} \)
17 \( 1 - 1.12e5T + 6.97e9T^{2} \)
19 \( 1 + 6.21e4iT - 1.69e10T^{2} \)
23 \( 1 - 2.50e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.20e5T + 5.00e11T^{2} \)
31 \( 1 - 1.61e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.31e6T + 3.51e12T^{2} \)
41 \( 1 + 6.67e5T + 7.98e12T^{2} \)
43 \( 1 + 2.96e6iT - 1.16e13T^{2} \)
47 \( 1 + 3.08e6iT - 2.38e13T^{2} \)
53 \( 1 - 5.63e6T + 6.22e13T^{2} \)
59 \( 1 - 3.54e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.55e7T + 1.91e14T^{2} \)
67 \( 1 - 3.28e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.69e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.96e7T + 8.06e14T^{2} \)
79 \( 1 + 1.49e7iT - 1.51e15T^{2} \)
83 \( 1 + 7.32e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.30e7T + 3.93e15T^{2} \)
97 \( 1 - 1.35e8T + 7.83e15T^{2} \)
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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

−10.66028368044289463999898617170, −10.00860151462118851842444010926, −8.792171265552519667451959514231, −7.69756078549083147845807034239, −7.15719626719452969562081918289, −5.90493604151648014731334152868, −4.63488357324205951685322230496, −3.49908281628102224947356178330, −1.29049767452969121204491647342, −0.42691431641188568222585251206, 1.06885960398154795258716222410, 2.38763335666200897333321817997, 3.38459935251068030137484849060, 4.75416537540866400725402329999, 6.11259386137137441875414417992, 7.69613389618786797656315692893, 8.466389918955893321840580766509, 9.380518057832048107277497638658, 10.34939043358293886068515467298, 11.46356260014649789573480515178

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