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Properties

Label 2-72-24.11-c21-0-11
Degree $2$
Conductor $72$
Sign $-0.997 + 0.0756i$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−259. + 1.42e3i)2-s + (−1.96e6 − 7.38e5i)4-s + 3.78e7·5-s − 1.95e8i·7-s + (1.56e9 − 2.60e9i)8-s + (−9.80e9 + 5.39e10i)10-s − 4.29e10i·11-s − 7.25e10i·13-s + (2.79e11 + 5.07e10i)14-s + (3.30e12 + 2.89e12i)16-s + 8.33e12i·17-s − 2.11e13·19-s + (−7.42e13 − 2.79e13i)20-s + (6.11e13 + 1.11e13i)22-s − 4.93e13·23-s + ⋯
L(s)  = 1  + (−0.178 + 0.983i)2-s + (−0.935 − 0.352i)4-s + 1.73·5-s − 0.262i·7-s + (0.513 − 0.857i)8-s + (−0.310 + 1.70i)10-s − 0.499i·11-s − 0.146i·13-s + (0.257 + 0.0469i)14-s + (0.752 + 0.659i)16-s + 1.00i·17-s − 0.791·19-s + (−1.62 − 0.610i)20-s + (0.491 + 0.0893i)22-s − 0.248·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0756i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.997 + 0.0756i$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :21/2),\ -0.997 + 0.0756i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.050377926\)
\(L(\frac12)\) \(\approx\) \(1.050377926\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (259. - 1.42e3i)T \)
3 \( 1 \)
good5 \( 1 - 3.78e7T + 4.76e14T^{2} \)
7 \( 1 + 1.95e8iT - 5.58e17T^{2} \)
11 \( 1 + 4.29e10iT - 7.40e21T^{2} \)
13 \( 1 + 7.25e10iT - 2.47e23T^{2} \)
17 \( 1 - 8.33e12iT - 6.90e25T^{2} \)
19 \( 1 + 2.11e13T + 7.14e26T^{2} \)
23 \( 1 + 4.93e13T + 3.94e28T^{2} \)
29 \( 1 + 2.69e15T + 5.13e30T^{2} \)
31 \( 1 + 1.12e15iT - 2.08e31T^{2} \)
37 \( 1 + 3.66e16iT - 8.55e32T^{2} \)
41 \( 1 - 8.22e16iT - 7.38e33T^{2} \)
43 \( 1 + 2.81e17T + 2.00e34T^{2} \)
47 \( 1 + 3.37e17T + 1.30e35T^{2} \)
53 \( 1 + 7.67e17T + 1.62e36T^{2} \)
59 \( 1 - 5.94e18iT - 1.54e37T^{2} \)
61 \( 1 + 3.41e18iT - 3.10e37T^{2} \)
67 \( 1 - 5.71e18T + 2.22e38T^{2} \)
71 \( 1 - 1.23e19T + 7.52e38T^{2} \)
73 \( 1 - 3.13e19T + 1.34e39T^{2} \)
79 \( 1 - 8.71e19iT - 7.08e39T^{2} \)
83 \( 1 - 4.57e19iT - 1.99e40T^{2} \)
89 \( 1 - 1.01e20iT - 8.65e40T^{2} \)
97 \( 1 + 2.74e20T + 5.27e41T^{2} \)
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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

−10.79386865707242086687372931502, −9.957988408670668285961762446365, −9.087693486443220872367194782778, −8.076832736723055233124204920329, −6.65176657242186565973255061219, −5.97750305503886328223437456822, −5.14974312334589740493649927669, −3.76655951021318039371822114114, −2.11306951148855019675883572343, −1.15879014836107748813257903029, 0.18584549468493889062778281012, 1.60616074123287240734793551429, 2.09522406066112366907319938593, 3.15421892753426421429963051359, 4.70062423353263422455480051454, 5.55160955664793689844009387737, 6.82555011727563209590431752158, 8.465225188696183390155628272793, 9.498639631292489273413529008672, 9.977783875533085545463570909010

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