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Properties

Label 2-54-27.11-c8-0-1
Degree 22
Conductor 5454
Sign −0.681+0.731i-0.681 + 0.731i
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 + (−74.4 − 31.9i)3-s + (−22.2 − 126. i)4-s + (−303. + 833. i)5-s + (818. − 412. i)6-s + (−11.0 + 62.7i)7-s + (1.25e3 + 724. i)8-s + (4.52e3 + 4.75e3i)9-s + (−5.01e3 − 8.68e3i)10-s + (9.13e3 + 2.50e4i)11-s + (−2.37e3 + 1.00e4i)12-s + (1.85e4 − 1.55e4i)13-s + (−463. − 552. i)14-s + (4.91e4 − 5.23e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (−6.81e4 + 3.93e4i)17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.918 − 0.394i)3-s + (−0.0868 − 0.492i)4-s + (−0.485 + 1.33i)5-s + (0.631 − 0.318i)6-s + (−0.00461 + 0.0261i)7-s + (0.306 + 0.176i)8-s + (0.689 + 0.724i)9-s + (−0.501 − 0.868i)10-s + (0.623 + 1.71i)11-s + (−0.114 + 0.486i)12-s + (0.650 − 0.545i)13-s + (−0.0120 − 0.0143i)14-s + (0.971 − 1.03i)15-s + (−0.234 + 0.0855i)16-s + (−0.816 + 0.471i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.681+0.731i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.681+0.731i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.681+0.731i-0.681 + 0.731i
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), −0.681+0.731i)(2,\ 54,\ (\ :4),\ -0.681 + 0.731i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.134806−0.310011i0.134806 - 0.310011i
L(12)L(\frac12) ≈\approx 0.134806−0.310011i0.134806 - 0.310011i
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+(74.4+31.9i)T 1 + (74.4 + 31.9i)T
good5 1+(303.−833.i)T+(−2.99e5−2.51e5i)T2 1 + (303. - 833. i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(11.0−62.7i)T+(−5.41e6−1.97e6i)T2 1 + (11.0 - 62.7i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−9.13e3−2.50e4i)T+(−1.64e8+1.37e8i)T2 1 + (-9.13e3 - 2.50e4i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(−1.85e4+1.55e4i)T+(1.41e8−8.03e8i)T2 1 + (-1.85e4 + 1.55e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(6.81e4−3.93e4i)T+(3.48e9−6.04e9i)T2 1 + (6.81e4 - 3.93e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(7.69e4−1.33e5i)T+(−8.49e9−1.47e10i)T2 1 + (7.69e4 - 1.33e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(5.40e4−9.53e3i)T+(7.35e10−2.67e10i)T2 1 + (5.40e4 - 9.53e3i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(1.55e4−1.84e4i)T+(−8.68e10−4.92e11i)T2 1 + (1.55e4 - 1.84e4i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(2.84e5+1.61e6i)T+(−8.01e11+2.91e11i)T2 1 + (2.84e5 + 1.61e6i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−5.91e5−1.02e6i)T+(−1.75e12+3.04e12i)T2 1 + (-5.91e5 - 1.02e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(−6.08e4−7.25e4i)T+(−1.38e12+7.86e12i)T2 1 + (-6.08e4 - 7.25e4i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(2.97e6−1.08e6i)T+(8.95e12−7.51e12i)T2 1 + (2.97e6 - 1.08e6i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(5.66e6+9.99e5i)T+(2.23e13+8.14e12i)T2 1 + (5.66e6 + 9.99e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+7.83e6iT−6.22e13T2 1 + 7.83e6iT - 6.22e13T^{2}
59 1+(6.93e6−1.90e7i)T+(−1.12e14−9.43e13i)T2 1 + (6.93e6 - 1.90e7i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(−1.42e5+8.06e5i)T+(−1.80e14−6.55e13i)T2 1 + (-1.42e5 + 8.06e5i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(9.66e6−8.11e6i)T+(7.05e13−3.99e14i)T2 1 + (9.66e6 - 8.11e6i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(−3.63e7+2.09e7i)T+(3.22e14−5.59e14i)T2 1 + (-3.63e7 + 2.09e7i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(8.83e6−1.53e7i)T+(−4.03e14−6.98e14i)T2 1 + (8.83e6 - 1.53e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(3.43e7+2.88e7i)T+(2.63e14+1.49e15i)T2 1 + (3.43e7 + 2.88e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−4.76e7+5.67e7i)T+(−3.91e14−2.21e15i)T2 1 + (-4.76e7 + 5.67e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(−3.61e7−2.08e7i)T+(1.96e15+3.40e15i)T2 1 + (-3.61e7 - 2.08e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(5.20e7−1.89e7i)T+(6.00e15−5.03e15i)T2 1 + (5.20e7 - 1.89e7i)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

−14.82436536019374369458988981074, −13.16372772139312463138006678858, −11.85583991857964082672043466852, −10.80488921783851465393497293736, −9.918648637296999513417484302853, −7.920258952094509600995421562661, −6.92642098978640571805927976962, −6.10638679797207151165062898788, −4.22101104096222851679562106197, −1.85076288852598622546265636855, 0.18405185792382956737479655291, 1.13193427113882348198192896499, 3.75936376892411747961505169905, 4.94305027211996878129861219755, 6.53869086288501055683229102861, 8.585476618130151297657613328505, 9.149174509344524948012225118259, 10.92043719664393673942089240707, 11.52315687226221980210961285274, 12.58244218387580028665557484845

Graph of the ZZ-function along the critical line