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Properties

Label 2-18-9.2-c10-0-9
Degree 22
Conductor 1818
Sign −0.186+0.982i-0.186 + 0.982i
Analytic cond. 11.436411.4364
Root an. cond. 3.381773.38177
Motivic weight 1010
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.5 + 11.3i)2-s + (1.63 − 242. i)3-s + (255. + 443. i)4-s + (−1.94e3 + 1.12e3i)5-s + (2.78e3 − 4.74e3i)6-s + (1.48e4 − 2.57e4i)7-s + 1.15e4i·8-s + (−5.90e4 − 796. i)9-s − 5.08e4·10-s + (−1.39e5 − 8.06e4i)11-s + (1.08e5 − 6.14e4i)12-s + (−2.49e5 − 4.31e5i)13-s + (5.82e5 − 3.36e5i)14-s + (2.69e5 + 4.74e5i)15-s + (−1.31e5 + 2.27e5i)16-s − 1.34e6i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.00674 − 0.999i)3-s + (0.249 + 0.433i)4-s + (−0.623 + 0.359i)5-s + (0.357 − 0.609i)6-s + (0.884 − 1.53i)7-s + 0.353i·8-s + (−0.999 − 0.0134i)9-s − 0.508·10-s + (−0.867 − 0.500i)11-s + (0.434 − 0.247i)12-s + (−0.671 − 1.16i)13-s + (1.08 − 0.625i)14-s + (0.355 + 0.625i)15-s + (−0.125 + 0.216i)16-s − 0.947i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.186+0.982i) Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+5) L(s)=((−0.186+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.186+0.982i-0.186 + 0.982i
Analytic conductor: 11.436411.4364
Root analytic conductor: 3.381773.38177
Motivic weight: 1010
Rational: no
Arithmetic: yes
Character: χ18(11,⋅)\chi_{18} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :5), −0.186+0.982i)(2,\ 18,\ (\ :5),\ -0.186 + 0.982i)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 1.20365−1.45427i1.20365 - 1.45427i
L(12)L(\frac12) ≈\approx 1.20365−1.45427i1.20365 - 1.45427i
L(6)L(6) 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+(−19.5−11.3i)T 1 + (-19.5 - 11.3i)T
3 1+(−1.63+242.i)T 1 + (-1.63 + 242. i)T
good5 1+(1.94e3−1.12e3i)T+(4.88e6−8.45e6i)T2 1 + (1.94e3 - 1.12e3i)T + (4.88e6 - 8.45e6i)T^{2}
7 1+(−1.48e4+2.57e4i)T+(−1.41e8−2.44e8i)T2 1 + (-1.48e4 + 2.57e4i)T + (-1.41e8 - 2.44e8i)T^{2}
11 1+(1.39e5+8.06e4i)T+(1.29e10+2.24e10i)T2 1 + (1.39e5 + 8.06e4i)T + (1.29e10 + 2.24e10i)T^{2}
13 1+(2.49e5+4.31e5i)T+(−6.89e10+1.19e11i)T2 1 + (2.49e5 + 4.31e5i)T + (-6.89e10 + 1.19e11i)T^{2}
17 1+1.34e6iT−2.01e12T2 1 + 1.34e6iT - 2.01e12T^{2}
19 1−4.27e6T+6.13e12T2 1 - 4.27e6T + 6.13e12T^{2}
23 1+(3.03e6−1.75e6i)T+(2.07e13−3.58e13i)T2 1 + (3.03e6 - 1.75e6i)T + (2.07e13 - 3.58e13i)T^{2}
29 1+(−1.44e7−8.32e6i)T+(2.10e14+3.64e14i)T2 1 + (-1.44e7 - 8.32e6i)T + (2.10e14 + 3.64e14i)T^{2}
31 1+(−5.30e6−9.19e6i)T+(−4.09e14+7.09e14i)T2 1 + (-5.30e6 - 9.19e6i)T + (-4.09e14 + 7.09e14i)T^{2}
37 1+3.61e6T+4.80e15T2 1 + 3.61e6T + 4.80e15T^{2}
41 1+(−1.48e7+8.59e6i)T+(6.71e15−1.16e16i)T2 1 + (-1.48e7 + 8.59e6i)T + (6.71e15 - 1.16e16i)T^{2}
43 1+(−7.92e7+1.37e8i)T+(−1.08e16−1.87e16i)T2 1 + (-7.92e7 + 1.37e8i)T + (-1.08e16 - 1.87e16i)T^{2}
47 1+(−1.61e8−9.32e7i)T+(2.62e16+4.55e16i)T2 1 + (-1.61e8 - 9.32e7i)T + (2.62e16 + 4.55e16i)T^{2}
53 1−6.84e8iT−1.74e17T2 1 - 6.84e8iT - 1.74e17T^{2}
59 1+(−1.08e8+6.28e7i)T+(2.55e17−4.42e17i)T2 1 + (-1.08e8 + 6.28e7i)T + (2.55e17 - 4.42e17i)T^{2}
61 1+(−3.64e8+6.31e8i)T+(−3.56e17−6.17e17i)T2 1 + (-3.64e8 + 6.31e8i)T + (-3.56e17 - 6.17e17i)T^{2}
67 1+(1.98e8+3.43e8i)T+(−9.11e17+1.57e18i)T2 1 + (1.98e8 + 3.43e8i)T + (-9.11e17 + 1.57e18i)T^{2}
71 1+2.21e9iT−3.25e18T2 1 + 2.21e9iT - 3.25e18T^{2}
73 1−6.41e8T+4.29e18T2 1 - 6.41e8T + 4.29e18T^{2}
79 1+(−1.46e9+2.54e9i)T+(−4.73e18−8.19e18i)T2 1 + (-1.46e9 + 2.54e9i)T + (-4.73e18 - 8.19e18i)T^{2}
83 1+(1.18e9+6.83e8i)T+(7.75e18+1.34e19i)T2 1 + (1.18e9 + 6.83e8i)T + (7.75e18 + 1.34e19i)T^{2}
89 1+6.25e9iT−3.11e19T2 1 + 6.25e9iT - 3.11e19T^{2}
97 1+(3.15e9−5.46e9i)T+(−3.68e19−6.38e19i)T2 1 + (3.15e9 - 5.46e9i)T + (-3.68e19 - 6.38e19i)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

−15.78840535997494407628359349298, −14.20102610660413620058260185962, −13.53259424899115549171101395133, −11.96625702267590143676207846735, −10.76649509259247461489694515507, −7.71361143278423195361047851821, −7.43018793220643174302704224712, −5.22233606885039852913381922554, −3.13776557197626920441650710966, −0.69650368064368702835346536604, 2.40006786794799974464692664715, 4.40666615693135088851119809481, 5.47234564483050704645684343990, 8.254974194779930402153405582165, 9.749557158527949614474353361951, 11.49370859838403023278828993746, 12.19167101216001842733549253895, 14.29742677141776892542059304385, 15.30082241642498593230167990206, 16.11808387804499566840224322251

Graph of the ZZ-function along the critical line