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Properties

Label 2-54-27.5-c8-0-23
Degree 22
Conductor 5454
Sign −0.686−0.727i-0.686 - 0.727i
Analytic cond. 21.998421.9984
Root an. cond. 4.690244.69024
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.27 + 8.66i)2-s + (−64.9 − 48.3i)3-s + (−22.2 + 126. i)4-s + (−403. − 1.10e3i)5-s + (−53.2 − 914. i)6-s + (−619. − 3.51e3i)7-s + (−1.25e3 + 724. i)8-s + (1.88e3 + 6.28e3i)9-s + (6.67e3 − 1.15e4i)10-s + (−7.11e3 + 1.95e4i)11-s + (7.54e3 − 7.11e3i)12-s + (2.16e4 + 1.81e4i)13-s + (2.59e4 − 3.09e4i)14-s + (−2.74e4 + 9.15e4i)15-s + (−1.53e4 − 5.60e3i)16-s + (−6.01e4 − 3.47e4i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−0.802 − 0.597i)3-s + (−0.0868 + 0.492i)4-s + (−0.645 − 1.77i)5-s + (−0.0411 − 0.705i)6-s + (−0.258 − 1.46i)7-s + (−0.306 + 0.176i)8-s + (0.286 + 0.958i)9-s + (0.667 − 1.15i)10-s + (−0.485 + 1.33i)11-s + (0.363 − 0.343i)12-s + (0.758 + 0.636i)13-s + (0.675 − 0.805i)14-s + (−0.541 + 1.80i)15-s + (−0.234 − 0.0855i)16-s + (−0.720 − 0.415i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.686−0.727i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.686−0.727i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.686−0.727i-0.686 - 0.727i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(5,⋅)\chi_{54} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), −0.686−0.727i)(2,\ 54,\ (\ :4),\ -0.686 - 0.727i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.0100124+0.0232097i0.0100124 + 0.0232097i
L(12)L(\frac12) ≈\approx 0.0100124+0.0232097i0.0100124 + 0.0232097i
L(5)L(5) 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+(−7.27−8.66i)T 1 + (-7.27 - 8.66i)T
3 1+(64.9+48.3i)T 1 + (64.9 + 48.3i)T
good5 1+(403.+1.10e3i)T+(−2.99e5+2.51e5i)T2 1 + (403. + 1.10e3i)T + (-2.99e5 + 2.51e5i)T^{2}
7 1+(619.+3.51e3i)T+(−5.41e6+1.97e6i)T2 1 + (619. + 3.51e3i)T + (-5.41e6 + 1.97e6i)T^{2}
11 1+(7.11e3−1.95e4i)T+(−1.64e8−1.37e8i)T2 1 + (7.11e3 - 1.95e4i)T + (-1.64e8 - 1.37e8i)T^{2}
13 1+(−2.16e4−1.81e4i)T+(1.41e8+8.03e8i)T2 1 + (-2.16e4 - 1.81e4i)T + (1.41e8 + 8.03e8i)T^{2}
17 1+(6.01e4+3.47e4i)T+(3.48e9+6.04e9i)T2 1 + (6.01e4 + 3.47e4i)T + (3.48e9 + 6.04e9i)T^{2}
19 1+(7.33e4+1.27e5i)T+(−8.49e9+1.47e10i)T2 1 + (7.33e4 + 1.27e5i)T + (-8.49e9 + 1.47e10i)T^{2}
23 1+(−3.07e5−5.42e4i)T+(7.35e10+2.67e10i)T2 1 + (-3.07e5 - 5.42e4i)T + (7.35e10 + 2.67e10i)T^{2}
29 1+(−2.37e5−2.83e5i)T+(−8.68e10+4.92e11i)T2 1 + (-2.37e5 - 2.83e5i)T + (-8.68e10 + 4.92e11i)T^{2}
31 1+(8.74e4−4.95e5i)T+(−8.01e11−2.91e11i)T2 1 + (8.74e4 - 4.95e5i)T + (-8.01e11 - 2.91e11i)T^{2}
37 1+(1.45e5−2.52e5i)T+(−1.75e12−3.04e12i)T2 1 + (1.45e5 - 2.52e5i)T + (-1.75e12 - 3.04e12i)T^{2}
41 1+(2.34e5−2.78e5i)T+(−1.38e12−7.86e12i)T2 1 + (2.34e5 - 2.78e5i)T + (-1.38e12 - 7.86e12i)T^{2}
43 1+(−8.71e5−3.17e5i)T+(8.95e12+7.51e12i)T2 1 + (-8.71e5 - 3.17e5i)T + (8.95e12 + 7.51e12i)T^{2}
47 1+(3.13e6−5.52e5i)T+(2.23e13−8.14e12i)T2 1 + (3.13e6 - 5.52e5i)T + (2.23e13 - 8.14e12i)T^{2}
53 1−1.14e7iT−6.22e13T2 1 - 1.14e7iT - 6.22e13T^{2}
59 1+(−4.51e6−1.24e7i)T+(−1.12e14+9.43e13i)T2 1 + (-4.51e6 - 1.24e7i)T + (-1.12e14 + 9.43e13i)T^{2}
61 1+(2.27e6+1.29e7i)T+(−1.80e14+6.55e13i)T2 1 + (2.27e6 + 1.29e7i)T + (-1.80e14 + 6.55e13i)T^{2}
67 1+(2.05e7+1.72e7i)T+(7.05e13+3.99e14i)T2 1 + (2.05e7 + 1.72e7i)T + (7.05e13 + 3.99e14i)T^{2}
71 1+(1.61e7+9.30e6i)T+(3.22e14+5.59e14i)T2 1 + (1.61e7 + 9.30e6i)T + (3.22e14 + 5.59e14i)T^{2}
73 1+(1.09e7+1.88e7i)T+(−4.03e14+6.98e14i)T2 1 + (1.09e7 + 1.88e7i)T + (-4.03e14 + 6.98e14i)T^{2}
79 1+(−1.58e7+1.32e7i)T+(2.63e14−1.49e15i)T2 1 + (-1.58e7 + 1.32e7i)T + (2.63e14 - 1.49e15i)T^{2}
83 1+(−8.05e6−9.59e6i)T+(−3.91e14+2.21e15i)T2 1 + (-8.05e6 - 9.59e6i)T + (-3.91e14 + 2.21e15i)T^{2}
89 1+(6.09e7−3.51e7i)T+(1.96e15−3.40e15i)T2 1 + (6.09e7 - 3.51e7i)T + (1.96e15 - 3.40e15i)T^{2}
97 1+(−1.24e8−4.51e7i)T+(6.00e15+5.03e15i)T2 1 + (-1.24e8 - 4.51e7i)T + (6.00e15 + 5.03e15i)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

−13.08336855283686089514075160669, −12.12807791239798743859871743053, −10.87905791056400884861345450932, −9.042036964775354111100947377047, −7.62678221106786775192040328365, −6.78671387056670746715558530377, −4.87691505077955818057332095276, −4.38009965338847418580295982424, −1.25347321485589206091544561974, −0.009197256914729123458217072580, 2.75251222111938434454976196780, 3.68723444492281639498389011225, 5.70821418162235409549246706872, 6.40545472165208942360285788294, 8.529288952899115962807859998501, 10.26394548494059577608444547265, 11.05018929318321842079488666875, 11.66548464758184046853709474614, 12.99916048857715579200992818514, 14.65052764151048346521851410340

Graph of the ZZ-function along the critical line