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Properties

Label 2-18-9.2-c6-0-4
Degree 22
Conductor 1818
Sign 0.435+0.900i0.435 + 0.900i
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 + (25.6 − 8.30i)3-s + (15.9 + 27.7i)4-s + (−9.39 + 5.42i)5-s + (−149. − 32.0i)6-s + (322. − 557. i)7-s − 181. i·8-s + (591. − 426. i)9-s + 61.3·10-s + (−1.05e3 − 611. i)11-s + (641. + 579. i)12-s + (1.16e3 + 2.01e3i)13-s + (−3.15e3 + 1.82e3i)14-s + (−196. + 217. i)15-s + (−512. + 886. i)16-s + 3.38e3i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.951 − 0.307i)3-s + (0.249 + 0.433i)4-s + (−0.0751 + 0.0433i)5-s + (−0.691 − 0.148i)6-s + (0.938 − 1.62i)7-s − 0.353i·8-s + (0.810 − 0.585i)9-s + 0.0613·10-s + (−0.795 − 0.459i)11-s + (0.371 + 0.335i)12-s + (0.529 + 0.917i)13-s + (−1.14 + 0.663i)14-s + (−0.0581 + 0.0643i)15-s + (−0.125 + 0.216i)16-s + 0.688i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((0.435+0.900i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+3) L(s)=((0.435+0.900i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: 0.435+0.900i0.435 + 0.900i
Analytic conductor: 4.140974.14097
Root analytic conductor: 2.034932.03493
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ18(11,⋅)\chi_{18} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :3), 0.435+0.900i)(2,\ 18,\ (\ :3),\ 0.435 + 0.900i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.30134−0.816198i1.30134 - 0.816198i
L(12)L(\frac12) ≈\approx 1.30134−0.816198i1.30134 - 0.816198i
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+(−25.6+8.30i)T 1 + (-25.6 + 8.30i)T
good5 1+(9.39−5.42i)T+(7.81e3−1.35e4i)T2 1 + (9.39 - 5.42i)T + (7.81e3 - 1.35e4i)T^{2}
7 1+(−322.+557.i)T+(−5.88e4−1.01e5i)T2 1 + (-322. + 557. i)T + (-5.88e4 - 1.01e5i)T^{2}
11 1+(1.05e3+611.i)T+(8.85e5+1.53e6i)T2 1 + (1.05e3 + 611. i)T + (8.85e5 + 1.53e6i)T^{2}
13 1+(−1.16e3−2.01e3i)T+(−2.41e6+4.18e6i)T2 1 + (-1.16e3 - 2.01e3i)T + (-2.41e6 + 4.18e6i)T^{2}
17 1−3.38e3iT−2.41e7T2 1 - 3.38e3iT - 2.41e7T^{2}
19 1−6.23e3T+4.70e7T2 1 - 6.23e3T + 4.70e7T^{2}
23 1+(1.24e4−7.18e3i)T+(7.40e7−1.28e8i)T2 1 + (1.24e4 - 7.18e3i)T + (7.40e7 - 1.28e8i)T^{2}
29 1+(−1.17e4−6.78e3i)T+(2.97e8+5.15e8i)T2 1 + (-1.17e4 - 6.78e3i)T + (2.97e8 + 5.15e8i)T^{2}
31 1+(4.39e3+7.60e3i)T+(−4.43e8+7.68e8i)T2 1 + (4.39e3 + 7.60e3i)T + (-4.43e8 + 7.68e8i)T^{2}
37 1+2.78e4T+2.56e9T2 1 + 2.78e4T + 2.56e9T^{2}
41 1+(4.79e4−2.76e4i)T+(2.37e9−4.11e9i)T2 1 + (4.79e4 - 2.76e4i)T + (2.37e9 - 4.11e9i)T^{2}
43 1+(2.69e4−4.66e4i)T+(−3.16e9−5.47e9i)T2 1 + (2.69e4 - 4.66e4i)T + (-3.16e9 - 5.47e9i)T^{2}
47 1+(−1.52e5−8.80e4i)T+(5.38e9+9.33e9i)T2 1 + (-1.52e5 - 8.80e4i)T + (5.38e9 + 9.33e9i)T^{2}
53 1+8.39e4iT−2.21e10T2 1 + 8.39e4iT - 2.21e10T^{2}
59 1+(−3.05e5+1.76e5i)T+(2.10e10−3.65e10i)T2 1 + (-3.05e5 + 1.76e5i)T + (2.10e10 - 3.65e10i)T^{2}
61 1+(8.33e4−1.44e5i)T+(−2.57e10−4.46e10i)T2 1 + (8.33e4 - 1.44e5i)T + (-2.57e10 - 4.46e10i)T^{2}
67 1+(−4.52e4−7.83e4i)T+(−4.52e10+7.83e10i)T2 1 + (-4.52e4 - 7.83e4i)T + (-4.52e10 + 7.83e10i)T^{2}
71 1−4.29e5iT−1.28e11T2 1 - 4.29e5iT - 1.28e11T^{2}
73 1−2.74e5T+1.51e11T2 1 - 2.74e5T + 1.51e11T^{2}
79 1+(1.63e5−2.82e5i)T+(−1.21e11−2.10e11i)T2 1 + (1.63e5 - 2.82e5i)T + (-1.21e11 - 2.10e11i)T^{2}
83 1+(8.58e5+4.95e5i)T+(1.63e11+2.83e11i)T2 1 + (8.58e5 + 4.95e5i)T + (1.63e11 + 2.83e11i)T^{2}
89 1+5.38e5iT−4.96e11T2 1 + 5.38e5iT - 4.96e11T^{2}
97 1+(−4.29e4+7.44e4i)T+(−4.16e11−7.21e11i)T2 1 + (-4.29e4 + 7.44e4i)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

−17.39362139472225840713865577452, −15.94984681057312767040374797794, −14.19766451854885609312429074702, −13.35437430747913765655275502238, −11.37167682607086716879820466026, −10.04134438017136674673503354559, −8.301251528714429114827107632503, −7.30463858938699683081063075740, −3.82220532046763172225376006567, −1.42489293160621109759278876731, 2.36328460336228856892955483412, 5.29840352800090307521949783263, 7.86653001350429968364917711616, 8.755304089846000449083736004119, 10.24405208308614589571577438940, 12.09364499384252336866868448324, 13.99120802944588996407320190901, 15.34613871577669982270178831846, 15.80690421979813048849326903126, 18.05464821131698736934918990294

Graph of the ZZ-function along the critical line