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Properties

Label 2-18-9.2-c6-0-1
Degree 22
Conductor 1818
Sign −0.738−0.673i-0.738 - 0.673i
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 + (−23.6 + 13.0i)3-s + (15.9 + 27.7i)4-s + (−95.8 + 55.3i)5-s + (−152. − 3.16i)6-s + (−163. + 282. i)7-s + 181. i·8-s + (390. − 615. i)9-s − 626.·10-s + (541. + 312. i)11-s + (−739. − 447. i)12-s + (1.01e3 + 1.75e3i)13-s + (−1.59e3 + 923. i)14-s + (1.54e3 − 2.55e3i)15-s + (−512. + 886. i)16-s − 4.12e3i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.876 + 0.481i)3-s + (0.249 + 0.433i)4-s + (−0.766 + 0.442i)5-s + (−0.706 − 0.0146i)6-s + (−0.475 + 0.823i)7-s + 0.353i·8-s + (0.535 − 0.844i)9-s − 0.626·10-s + (0.407 + 0.235i)11-s + (−0.427 − 0.258i)12-s + (0.461 + 0.799i)13-s + (−0.582 + 0.336i)14-s + (0.458 − 0.757i)15-s + (−0.125 + 0.216i)16-s − 0.839i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.738−0.673i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.673i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+3) L(s)=((−0.738−0.673i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.738−0.673i-0.738 - 0.673i
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), −0.738−0.673i)(2,\ 18,\ (\ :3),\ -0.738 - 0.673i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.421310+1.08697i0.421310 + 1.08697i
L(12)L(\frac12) ≈\approx 0.421310+1.08697i0.421310 + 1.08697i
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+(23.6−13.0i)T 1 + (23.6 - 13.0i)T
good5 1+(95.8−55.3i)T+(7.81e3−1.35e4i)T2 1 + (95.8 - 55.3i)T + (7.81e3 - 1.35e4i)T^{2}
7 1+(163.−282.i)T+(−5.88e4−1.01e5i)T2 1 + (163. - 282. i)T + (-5.88e4 - 1.01e5i)T^{2}
11 1+(−541.−312.i)T+(8.85e5+1.53e6i)T2 1 + (-541. - 312. i)T + (8.85e5 + 1.53e6i)T^{2}
13 1+(−1.01e3−1.75e3i)T+(−2.41e6+4.18e6i)T2 1 + (-1.01e3 - 1.75e3i)T + (-2.41e6 + 4.18e6i)T^{2}
17 1+4.12e3iT−2.41e7T2 1 + 4.12e3iT - 2.41e7T^{2}
19 1−1.21e4T+4.70e7T2 1 - 1.21e4T + 4.70e7T^{2}
23 1+(1.85e4−1.07e4i)T+(7.40e7−1.28e8i)T2 1 + (1.85e4 - 1.07e4i)T + (7.40e7 - 1.28e8i)T^{2}
29 1+(2.14e4+1.23e4i)T+(2.97e8+5.15e8i)T2 1 + (2.14e4 + 1.23e4i)T + (2.97e8 + 5.15e8i)T^{2}
31 1+(−2.02e4−3.51e4i)T+(−4.43e8+7.68e8i)T2 1 + (-2.02e4 - 3.51e4i)T + (-4.43e8 + 7.68e8i)T^{2}
37 1+3.00e4T+2.56e9T2 1 + 3.00e4T + 2.56e9T^{2}
41 1+(−5.12e4+2.96e4i)T+(2.37e9−4.11e9i)T2 1 + (-5.12e4 + 2.96e4i)T + (2.37e9 - 4.11e9i)T^{2}
43 1+(−4.39e4+7.61e4i)T+(−3.16e9−5.47e9i)T2 1 + (-4.39e4 + 7.61e4i)T + (-3.16e9 - 5.47e9i)T^{2}
47 1+(−1.12e5−6.48e4i)T+(5.38e9+9.33e9i)T2 1 + (-1.12e5 - 6.48e4i)T + (5.38e9 + 9.33e9i)T^{2}
53 1+1.65e5iT−2.21e10T2 1 + 1.65e5iT - 2.21e10T^{2}
59 1+(−4.61e4+2.66e4i)T+(2.10e10−3.65e10i)T2 1 + (-4.61e4 + 2.66e4i)T + (2.10e10 - 3.65e10i)T^{2}
61 1+(−1.33e5+2.31e5i)T+(−2.57e10−4.46e10i)T2 1 + (-1.33e5 + 2.31e5i)T + (-2.57e10 - 4.46e10i)T^{2}
67 1+(−2.03e5−3.52e5i)T+(−4.52e10+7.83e10i)T2 1 + (-2.03e5 - 3.52e5i)T + (-4.52e10 + 7.83e10i)T^{2}
71 1−1.86e5iT−1.28e11T2 1 - 1.86e5iT - 1.28e11T^{2}
73 1+2.42e5T+1.51e11T2 1 + 2.42e5T + 1.51e11T^{2}
79 1+(−6.31e4+1.09e5i)T+(−1.21e11−2.10e11i)T2 1 + (-6.31e4 + 1.09e5i)T + (-1.21e11 - 2.10e11i)T^{2}
83 1+(5.97e4+3.44e4i)T+(1.63e11+2.83e11i)T2 1 + (5.97e4 + 3.44e4i)T + (1.63e11 + 2.83e11i)T^{2}
89 1−4.13e5iT−4.96e11T2 1 - 4.13e5iT - 4.96e11T^{2}
97 1+(4.89e5−8.47e5i)T+(−4.16e11−7.21e11i)T2 1 + (4.89e5 - 8.47e5i)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

−17.69789604301638132233114662574, −15.93907958824787144746024713795, −15.74543957640731671612848904290, −14.06070429600773594131927022633, −12.08647730904914928012726851240, −11.50432953680973197061555607137, −9.478926970833758556453354135415, −7.14167281862408199451551996203, −5.61324880855561358044482459906, −3.73828622670048338695051379995, 0.75577935722495958755642441221, 4.02785149678213105018757182993, 5.95698820449372778167748373641, 7.69664809858457341670088372738, 10.29121891108836585047701171291, 11.61459622390617229864328116969, 12.65553618662750616231382745911, 13.81601456177163850825466937850, 15.77217395051018090421217787271, 16.68731846217994426617589859863

Graph of the ZZ-function along the critical line