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Properties

Label 2-63-9.4-c7-0-14
Degree 22
Conductor 6363
Sign −0.157+0.987i-0.157 + 0.987i
Analytic cond. 19.680219.6802
Root an. cond. 4.436244.43624
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.42 + 14.5i)2-s + (−7.74 + 46.1i)3-s + (−77.8 + 134. i)4-s + (−239. + 413. i)5-s + (−738. + 275. i)6-s + (−171.5 − 297. i)7-s − 467.·8-s + (−2.06e3 − 714. i)9-s − 8.05e3·10-s + (3.72e3 + 6.45e3i)11-s + (−5.61e3 − 4.63e3i)12-s + (5.03e3 − 8.72e3i)13-s + (2.88e3 − 5.00e3i)14-s + (−1.72e4 − 1.42e4i)15-s + (6.03e3 + 1.04e4i)16-s + 3.59e3·17-s + ⋯
L(s)  = 1  + (0.744 + 1.28i)2-s + (−0.165 + 0.986i)3-s + (−0.608 + 1.05i)4-s + (−0.855 + 1.48i)5-s + (−1.39 + 0.520i)6-s + (−0.188 − 0.327i)7-s − 0.322·8-s + (−0.945 − 0.326i)9-s − 2.54·10-s + (0.844 + 1.46i)11-s + (−0.938 − 0.774i)12-s + (0.636 − 1.10i)13-s + (0.281 − 0.487i)14-s + (−1.31 − 1.08i)15-s + (0.368 + 0.637i)16-s + 0.177·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.157+0.987i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((−0.157+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.157+0.987i-0.157 + 0.987i
Analytic conductor: 19.680219.6802
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ63(22,⋅)\chi_{63} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :7/2), −0.157+0.987i)(2,\ 63,\ (\ :7/2),\ -0.157 + 0.987i)

Particular Values

L(4)L(4) ≈\approx 1.24736−1.46246i1.24736 - 1.46246i
L(12)L(\frac12) ≈\approx 1.24736−1.46246i1.24736 - 1.46246i
L(92)L(\frac{9}{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)
bad3 1+(7.74−46.1i)T 1 + (7.74 - 46.1i)T
7 1+(171.5+297.i)T 1 + (171.5 + 297. i)T
good2 1+(−8.42−14.5i)T+(−64+110.i)T2 1 + (-8.42 - 14.5i)T + (-64 + 110. i)T^{2}
5 1+(239.−413.i)T+(−3.90e4−6.76e4i)T2 1 + (239. - 413. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−3.72e3−6.45e3i)T+(−9.74e6+1.68e7i)T2 1 + (-3.72e3 - 6.45e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−5.03e3+8.72e3i)T+(−3.13e7−5.43e7i)T2 1 + (-5.03e3 + 8.72e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1−3.59e3T+4.10e8T2 1 - 3.59e3T + 4.10e8T^{2}
19 1−1.38e4T+8.93e8T2 1 - 1.38e4T + 8.93e8T^{2}
23 1+(4.10e4−7.10e4i)T+(−1.70e9−2.94e9i)T2 1 + (4.10e4 - 7.10e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−1.17e4−2.04e4i)T+(−8.62e9+1.49e10i)T2 1 + (-1.17e4 - 2.04e4i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(−8.13e4+1.40e5i)T+(−1.37e10−2.38e10i)T2 1 + (-8.13e4 + 1.40e5i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1+2.98e4T+9.49e10T2 1 + 2.98e4T + 9.49e10T^{2}
41 1+(−1.16e5+2.01e5i)T+(−9.73e10−1.68e11i)T2 1 + (-1.16e5 + 2.01e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(2.83e5+4.91e5i)T+(−1.35e11+2.35e11i)T2 1 + (2.83e5 + 4.91e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(−2.58e5−4.47e5i)T+(−2.53e11+4.38e11i)T2 1 + (-2.58e5 - 4.47e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+8.89e5T+1.17e12T2 1 + 8.89e5T + 1.17e12T^{2}
59 1+(7.91e5−1.37e6i)T+(−1.24e12−2.15e12i)T2 1 + (7.91e5 - 1.37e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(6.63e5+1.14e6i)T+(−1.57e12+2.72e12i)T2 1 + (6.63e5 + 1.14e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(2.04e6−3.53e6i)T+(−3.03e12−5.24e12i)T2 1 + (2.04e6 - 3.53e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+2.03e6T+9.09e12T2 1 + 2.03e6T + 9.09e12T^{2}
73 1−4.81e6T+1.10e13T2 1 - 4.81e6T + 1.10e13T^{2}
79 1+(−4.01e6−6.95e6i)T+(−9.60e12+1.66e13i)T2 1 + (-4.01e6 - 6.95e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(1.44e6+2.50e6i)T+(−1.35e13+2.35e13i)T2 1 + (1.44e6 + 2.50e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1+2.96e6T+4.42e13T2 1 + 2.96e6T + 4.42e13T^{2}
97 1+(−1.63e6−2.82e6i)T+(−4.03e13+6.99e13i)T2 1 + (-1.63e6 - 2.82e6i)T + (-4.03e13 + 6.99e13i)T^{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

−14.70000186519367456942184928302, −13.80423858309824188501461370378, −12.07249826597750461192293943815, −10.84986338005258066164917288614, −9.843968409373144213514078509609, −7.88613125038514239445831640760, −6.98100867006250586048144838472, −5.82520062591746039855394579683, −4.25139900951174705139294249146, −3.41370728299643859441811378459, 0.60260249058226886645683524577, 1.53213822784396635262681790047, 3.39310897971547671604669763617, 4.69027830169018431413242429769, 6.18694451917035306845775326691, 8.166172949553270206501120403384, 9.054811063991160473773022710369, 11.14649055344862276126221962047, 11.88736439631267706541083337176, 12.38436984610783824300938697760

Graph of the ZZ-function along the critical line