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Properties

Label 2-72-24.11-c21-0-11
Degree 22
Conductor 7272
Sign −0.997+0.0756i-0.997 + 0.0756i
Analytic cond. 201.223201.223
Root an. cond. 14.185314.1853
Motivic weight 2121
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.997+0.0756i) Λ‾(22−s)\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}
Λ(s)=(72s/2 ΓC(s+21/2) L(s)=((−0.997+0.0756i) Λ‾(1−s)\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: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.997+0.0756i-0.997 + 0.0756i
Analytic conductor: 201.223201.223
Root analytic conductor: 14.185314.1853
Motivic weight: 2121
Rational: no
Arithmetic: yes
Character: χ72(35,⋅)\chi_{72} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :21/2), −0.997+0.0756i)(2,\ 72,\ (\ :21/2),\ -0.997 + 0.0756i)

Particular Values

L(11)L(11) ≈\approx 1.0503779261.050377926
L(12)L(\frac12) ≈\approx 1.0503779261.050377926
L(232)L(\frac{23}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(259.−1.42e3i)T 1 + (259. - 1.42e3i)T
3 1 1
good5 1−3.78e7T+4.76e14T2 1 - 3.78e7T + 4.76e14T^{2}
7 1+1.95e8iT−5.58e17T2 1 + 1.95e8iT - 5.58e17T^{2}
11 1+4.29e10iT−7.40e21T2 1 + 4.29e10iT - 7.40e21T^{2}
13 1+7.25e10iT−2.47e23T2 1 + 7.25e10iT - 2.47e23T^{2}
17 1−8.33e12iT−6.90e25T2 1 - 8.33e12iT - 6.90e25T^{2}
19 1+2.11e13T+7.14e26T2 1 + 2.11e13T + 7.14e26T^{2}
23 1+4.93e13T+3.94e28T2 1 + 4.93e13T + 3.94e28T^{2}
29 1+2.69e15T+5.13e30T2 1 + 2.69e15T + 5.13e30T^{2}
31 1+1.12e15iT−2.08e31T2 1 + 1.12e15iT - 2.08e31T^{2}
37 1+3.66e16iT−8.55e32T2 1 + 3.66e16iT - 8.55e32T^{2}
41 1−8.22e16iT−7.38e33T2 1 - 8.22e16iT - 7.38e33T^{2}
43 1+2.81e17T+2.00e34T2 1 + 2.81e17T + 2.00e34T^{2}
47 1+3.37e17T+1.30e35T2 1 + 3.37e17T + 1.30e35T^{2}
53 1+7.67e17T+1.62e36T2 1 + 7.67e17T + 1.62e36T^{2}
59 1−5.94e18iT−1.54e37T2 1 - 5.94e18iT - 1.54e37T^{2}
61 1+3.41e18iT−3.10e37T2 1 + 3.41e18iT - 3.10e37T^{2}
67 1−5.71e18T+2.22e38T2 1 - 5.71e18T + 2.22e38T^{2}
71 1−1.23e19T+7.52e38T2 1 - 1.23e19T + 7.52e38T^{2}
73 1−3.13e19T+1.34e39T2 1 - 3.13e19T + 1.34e39T^{2}
79 1−8.71e19iT−7.08e39T2 1 - 8.71e19iT - 7.08e39T^{2}
83 1−4.57e19iT−1.99e40T2 1 - 4.57e19iT - 1.99e40T^{2}
89 1−1.01e20iT−8.65e40T2 1 - 1.01e20iT - 8.65e40T^{2}
97 1+2.74e20T+5.27e41T2 1 + 2.74e20T + 5.27e41T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line