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Properties

Label 2-72-8.5-c3-0-0
Degree $2$
Conductor $72$
Sign $-i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 8.00·4-s + 19.7i·5-s − 34·7-s + 22.6i·8-s + 56.0·10-s − 5.65i·11-s + 96.1i·14-s + 64.0·16-s − 158. i·20-s − 16.0·22-s − 267·25-s + 272.·28-s + 223. i·29-s − 70·31-s − 181. i·32-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 1.77i·5-s − 1.83·7-s + 1.00i·8-s + 1.77·10-s − 0.155i·11-s + 1.83i·14-s + 1.00·16-s − 1.77i·20-s − 0.155·22-s − 2.13·25-s + 1.83·28-s + 1.43i·29-s − 0.405·31-s − 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.405861 + 0.405861i\)
\(L(\frac12)\) \(\approx\) \(0.405861 + 0.405861i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
3 \( 1 \)
good5 \( 1 - 19.7iT - 125T^{2} \)
7 \( 1 + 34T + 343T^{2} \)
11 \( 1 + 5.65iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 223. iT - 2.43e4T^{2} \)
31 \( 1 + 70T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 579. iT - 1.48e5T^{2} \)
59 \( 1 - 554. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 322T + 3.89e5T^{2} \)
79 \( 1 - 1.37e3T + 4.93e5T^{2} \)
83 \( 1 + 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 574T + 9.12e5T^{2} \)
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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

−14.17718607903619856227228895598, −13.24653749272695992031803724784, −12.16288556300628572989845008256, −10.85506024077384268326097553077, −10.19958919822051231880958415977, −9.177395630532827953833012070991, −7.20642234479338213022816148277, −6.04869878860156343370791160501, −3.58957451059288321917700364229, −2.77043211972590253310789402059, 0.36614719445429008901171108053, 3.95790739940280129625087322754, 5.37971887021428514516203029817, 6.54035295236445653841503674898, 8.083141718472667870314990299212, 9.237740399249454937352647202806, 9.807111819011910094861810777203, 12.25519550759253519001320441376, 12.97062261434933240882618161972, 13.61939056226155163799892746589

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