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Properties

Label 2-18-9.5-c6-0-3
Degree 22
Conductor 1818
Sign −0.629+0.777i-0.629 + 0.777i
Analytic cond. 4.140974.14097
Root an. cond. 2.034932.03493
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 + 2.82i)2-s + (−11.1 + 24.5i)3-s + (15.9 − 27.7i)4-s + (−39.5 − 22.8i)5-s + (−14.6 − 152. i)6-s + (−245. − 424. i)7-s + 181. i·8-s + (−478. − 550. i)9-s + 258.·10-s + (−873. + 504. i)11-s + (501. + 703. i)12-s + (466. − 808. i)13-s + (2.40e3 + 1.38e3i)14-s + (1.00e3 − 716. i)15-s + (−512. − 886. i)16-s + 8.09e3i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.414 + 0.909i)3-s + (0.249 − 0.433i)4-s + (−0.316 − 0.182i)5-s + (−0.0677 − 0.703i)6-s + (−0.714 − 1.23i)7-s + 0.353i·8-s + (−0.655 − 0.754i)9-s + 0.258·10-s + (−0.656 + 0.378i)11-s + (0.290 + 0.407i)12-s + (0.212 − 0.368i)13-s + (0.875 + 0.505i)14-s + (0.297 − 0.212i)15-s + (−0.125 − 0.216i)16-s + 1.64i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.629+0.777i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+3) L(s)=((−0.629+0.777i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.629+0.777i-0.629 + 0.777i
Analytic conductor: 4.140974.14097
Root analytic conductor: 2.034932.03493
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ18(5,⋅)\chi_{18} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :3), −0.629+0.777i)(2,\ 18,\ (\ :3),\ -0.629 + 0.777i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.0318399−0.0667645i0.0318399 - 0.0667645i
L(12)L(\frac12) ≈\approx 0.0318399−0.0667645i0.0318399 - 0.0667645i
L(4)L(4) 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+(4.89−2.82i)T 1 + (4.89 - 2.82i)T
3 1+(11.1−24.5i)T 1 + (11.1 - 24.5i)T
good5 1+(39.5+22.8i)T+(7.81e3+1.35e4i)T2 1 + (39.5 + 22.8i)T + (7.81e3 + 1.35e4i)T^{2}
7 1+(245.+424.i)T+(−5.88e4+1.01e5i)T2 1 + (245. + 424. i)T + (-5.88e4 + 1.01e5i)T^{2}
11 1+(873.−504.i)T+(8.85e5−1.53e6i)T2 1 + (873. - 504. i)T + (8.85e5 - 1.53e6i)T^{2}
13 1+(−466.+808.i)T+(−2.41e6−4.18e6i)T2 1 + (-466. + 808. i)T + (-2.41e6 - 4.18e6i)T^{2}
17 1−8.09e3iT−2.41e7T2 1 - 8.09e3iT - 2.41e7T^{2}
19 1+7.72e3T+4.70e7T2 1 + 7.72e3T + 4.70e7T^{2}
23 1+(1.18e4+6.84e3i)T+(7.40e7+1.28e8i)T2 1 + (1.18e4 + 6.84e3i)T + (7.40e7 + 1.28e8i)T^{2}
29 1+(1.96e3−1.13e3i)T+(2.97e8−5.15e8i)T2 1 + (1.96e3 - 1.13e3i)T + (2.97e8 - 5.15e8i)T^{2}
31 1+(1.70e4−2.95e4i)T+(−4.43e8−7.68e8i)T2 1 + (1.70e4 - 2.95e4i)T + (-4.43e8 - 7.68e8i)T^{2}
37 1−9.20e4T+2.56e9T2 1 - 9.20e4T + 2.56e9T^{2}
41 1+(3.10e4+1.79e4i)T+(2.37e9+4.11e9i)T2 1 + (3.10e4 + 1.79e4i)T + (2.37e9 + 4.11e9i)T^{2}
43 1+(−3.45e4−5.98e4i)T+(−3.16e9+5.47e9i)T2 1 + (-3.45e4 - 5.98e4i)T + (-3.16e9 + 5.47e9i)T^{2}
47 1+(−1.32e4+7.62e3i)T+(5.38e9−9.33e9i)T2 1 + (-1.32e4 + 7.62e3i)T + (5.38e9 - 9.33e9i)T^{2}
53 1+2.36e5iT−2.21e10T2 1 + 2.36e5iT - 2.21e10T^{2}
59 1+(2.21e5+1.28e5i)T+(2.10e10+3.65e10i)T2 1 + (2.21e5 + 1.28e5i)T + (2.10e10 + 3.65e10i)T^{2}
61 1+(1.99e4+3.45e4i)T+(−2.57e10+4.46e10i)T2 1 + (1.99e4 + 3.45e4i)T + (-2.57e10 + 4.46e10i)T^{2}
67 1+(1.60e5−2.77e5i)T+(−4.52e10−7.83e10i)T2 1 + (1.60e5 - 2.77e5i)T + (-4.52e10 - 7.83e10i)T^{2}
71 1−4.04e5iT−1.28e11T2 1 - 4.04e5iT - 1.28e11T^{2}
73 1−3.93e5T+1.51e11T2 1 - 3.93e5T + 1.51e11T^{2}
79 1+(4.49e5+7.78e5i)T+(−1.21e11+2.10e11i)T2 1 + (4.49e5 + 7.78e5i)T + (-1.21e11 + 2.10e11i)T^{2}
83 1+(−1.54e5+8.94e4i)T+(1.63e11−2.83e11i)T2 1 + (-1.54e5 + 8.94e4i)T + (1.63e11 - 2.83e11i)T^{2}
89 1−8.26e5iT−4.96e11T2 1 - 8.26e5iT - 4.96e11T^{2}
97 1+(3.17e5+5.50e5i)T+(−4.16e11+7.21e11i)T2 1 + (3.17e5 + 5.50e5i)T + (-4.16e11 + 7.21e11i)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

−16.75939012292823276147856828268, −15.93398224942121299746825444193, −14.69785449599607124064918014746, −12.77161868001907682824024231596, −10.73917007710819511588954036731, −10.04541249643037992735655091509, −8.208698560763989655009045311140, −6.28948962922804108803581722212, −4.11972677104944947336130834320, −0.05974136556983953401149895312, 2.51361554893894988813593547435, 5.99014583345743936276167598944, 7.67850870224326549487971886562, 9.244253685062874394402702866234, 11.20287774412839682591382636092, 12.19402719574377865925444801967, 13.44314677945778322562652032342, 15.54107748701911683921088762785, 16.70336866291349482255243499531, 18.39116073141736199074552062969

Graph of the ZZ-function along the critical line