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Properties

Label 2-54-27.11-c8-0-22
Degree 22
Conductor 5454
Sign −0.999−0.0297i-0.999 - 0.0297i
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 + (13.3 − 79.8i)3-s + (−22.2 − 126. i)4-s + (258. − 711. i)5-s + (−595. − 696. i)6-s + (690. − 3.91e3i)7-s + (−1.25e3 − 724. i)8-s + (−6.20e3 − 2.12e3i)9-s + (−4.28e3 − 7.41e3i)10-s + (9.36e3 + 2.57e4i)11-s + (−1.03e4 + 98.2i)12-s + (3.45e4 − 2.90e4i)13-s + (−2.89e4 − 3.44e4i)14-s + (−5.34e4 − 3.01e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (6.12e4 − 3.53e4i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.164 − 0.986i)3-s + (−0.0868 − 0.492i)4-s + (0.414 − 1.13i)5-s + (−0.459 − 0.537i)6-s + (0.287 − 1.63i)7-s + (−0.306 − 0.176i)8-s + (−0.946 − 0.324i)9-s + (−0.428 − 0.741i)10-s + (0.639 + 1.75i)11-s + (−0.499 + 0.00473i)12-s + (1.21 − 1.01i)13-s + (−0.753 − 0.897i)14-s + (−1.05 − 0.595i)15-s + (−0.234 + 0.0855i)16-s + (0.733 − 0.423i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.999−0.0297i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0297i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.999−0.0297i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.999 - 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.999−0.0297i-0.999 - 0.0297i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(11,⋅)\chi_{54} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), −0.999−0.0297i)(2,\ 54,\ (\ :4),\ -0.999 - 0.0297i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.0427853+2.87596i0.0427853 + 2.87596i
L(12)L(\frac12) ≈\approx 0.0427853+2.87596i0.0427853 + 2.87596i
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+(−13.3+79.8i)T 1 + (-13.3 + 79.8i)T
good5 1+(−258.+711.i)T+(−2.99e5−2.51e5i)T2 1 + (-258. + 711. i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(−690.+3.91e3i)T+(−5.41e6−1.97e6i)T2 1 + (-690. + 3.91e3i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−9.36e3−2.57e4i)T+(−1.64e8+1.37e8i)T2 1 + (-9.36e3 - 2.57e4i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(−3.45e4+2.90e4i)T+(1.41e8−8.03e8i)T2 1 + (-3.45e4 + 2.90e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(−6.12e4+3.53e4i)T+(3.48e9−6.04e9i)T2 1 + (-6.12e4 + 3.53e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(6.79e4−1.17e5i)T+(−8.49e9−1.47e10i)T2 1 + (6.79e4 - 1.17e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(−1.76e5+3.10e4i)T+(7.35e10−2.67e10i)T2 1 + (-1.76e5 + 3.10e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(5.77e5−6.88e5i)T+(−8.68e10−4.92e11i)T2 1 + (5.77e5 - 6.88e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(−3.66e4−2.07e5i)T+(−8.01e11+2.91e11i)T2 1 + (-3.66e4 - 2.07e5i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−6.81e5−1.18e6i)T+(−1.75e12+3.04e12i)T2 1 + (-6.81e5 - 1.18e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(2.76e5+3.30e5i)T+(−1.38e12+7.86e12i)T2 1 + (2.76e5 + 3.30e5i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(1.34e6−4.89e5i)T+(8.95e12−7.51e12i)T2 1 + (1.34e6 - 4.89e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(−4.35e6−7.68e5i)T+(2.23e13+8.14e12i)T2 1 + (-4.35e6 - 7.68e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+4.62e6iT−6.22e13T2 1 + 4.62e6iT - 6.22e13T^{2}
59 1+(−2.57e6+7.07e6i)T+(−1.12e14−9.43e13i)T2 1 + (-2.57e6 + 7.07e6i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(−4.16e4+2.36e5i)T+(−1.80e14−6.55e13i)T2 1 + (-4.16e4 + 2.36e5i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−1.91e7+1.60e7i)T+(7.05e13−3.99e14i)T2 1 + (-1.91e7 + 1.60e7i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(−3.71e7+2.14e7i)T+(3.22e14−5.59e14i)T2 1 + (-3.71e7 + 2.14e7i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(9.53e6−1.65e7i)T+(−4.03e14−6.98e14i)T2 1 + (9.53e6 - 1.65e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(−2.53e7−2.12e7i)T+(2.63e14+1.49e15i)T2 1 + (-2.53e7 - 2.12e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(4.49e7−5.35e7i)T+(−3.91e14−2.21e15i)T2 1 + (4.49e7 - 5.35e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(−4.16e7−2.40e7i)T+(1.96e15+3.40e15i)T2 1 + (-4.16e7 - 2.40e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(8.58e7−3.12e7i)T+(6.00e15−5.03e15i)T2 1 + (8.58e7 - 3.12e7i)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

−12.93726684131396316141496444337, −12.38342603769200019341270594650, −10.85899314911054724427007971285, −9.598493801347303789457466090013, −8.095853775550494426571685623140, −6.84051588143840199351203569342, −5.18914629074992648330219166831, −3.77006389383881436947125961238, −1.54835647217790519136127455683, −0.957292136891916134373912572726, 2.59945405410264058090911195694, 3.75399423217055683249051995156, 5.66471230609176311416989816907, 6.28229438871135467071267719029, 8.505775907881877060456757622076, 9.172730291572501689104624644138, 11.00884157098562948592571241199, 11.57664799288023082511738212406, 13.57229141477462694363171923569, 14.44206535102591722411654111248

Graph of the ZZ-function along the critical line