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Properties

Label 2-18-9.2-c6-0-5
Degree 22
Conductor 1818
Sign 0.840+0.541i0.840 + 0.541i
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 + (−14.9 − 22.4i)3-s + (15.9 + 27.7i)4-s + (202. − 116. i)5-s + (−9.58 − 152. i)6-s + (95.5 − 165. i)7-s + 181. i·8-s + (−282. + 672. i)9-s + 1.32e3·10-s + (−673. − 388. i)11-s + (384. − 773. i)12-s + (45.5 + 78.9i)13-s + (936. − 540. i)14-s + (−5.64e3 − 2.80e3i)15-s + (−512. + 886. i)16-s + 7.04e3i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.553 − 0.832i)3-s + (0.249 + 0.433i)4-s + (1.61 − 0.934i)5-s + (−0.0443 − 0.705i)6-s + (0.278 − 0.482i)7-s + 0.353i·8-s + (−0.387 + 0.921i)9-s + 1.32·10-s + (−0.505 − 0.291i)11-s + (0.222 − 0.447i)12-s + (0.0207 + 0.0359i)13-s + (0.341 − 0.197i)14-s + (−1.67 − 0.830i)15-s + (−0.125 + 0.216i)16-s + 1.43i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((0.840+0.541i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+3) L(s)=((0.840+0.541i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: 0.840+0.541i0.840 + 0.541i
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), 0.840+0.541i)(2,\ 18,\ (\ :3),\ 0.840 + 0.541i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.98741−0.584998i1.98741 - 0.584998i
L(12)L(\frac12) ≈\approx 1.98741−0.584998i1.98741 - 0.584998i
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+(14.9+22.4i)T 1 + (14.9 + 22.4i)T
good5 1+(−202.+116.i)T+(7.81e3−1.35e4i)T2 1 + (-202. + 116. i)T + (7.81e3 - 1.35e4i)T^{2}
7 1+(−95.5+165.i)T+(−5.88e4−1.01e5i)T2 1 + (-95.5 + 165. i)T + (-5.88e4 - 1.01e5i)T^{2}
11 1+(673.+388.i)T+(8.85e5+1.53e6i)T2 1 + (673. + 388. i)T + (8.85e5 + 1.53e6i)T^{2}
13 1+(−45.5−78.9i)T+(−2.41e6+4.18e6i)T2 1 + (-45.5 - 78.9i)T + (-2.41e6 + 4.18e6i)T^{2}
17 1−7.04e3iT−2.41e7T2 1 - 7.04e3iT - 2.41e7T^{2}
19 1−2.73e3T+4.70e7T2 1 - 2.73e3T + 4.70e7T^{2}
23 1+(1.72e4−9.94e3i)T+(7.40e7−1.28e8i)T2 1 + (1.72e4 - 9.94e3i)T + (7.40e7 - 1.28e8i)T^{2}
29 1+(−2.71e4−1.56e4i)T+(2.97e8+5.15e8i)T2 1 + (-2.71e4 - 1.56e4i)T + (2.97e8 + 5.15e8i)T^{2}
31 1+(−6.17e3−1.06e4i)T+(−4.43e8+7.68e8i)T2 1 + (-6.17e3 - 1.06e4i)T + (-4.43e8 + 7.68e8i)T^{2}
37 1+2.79e4T+2.56e9T2 1 + 2.79e4T + 2.56e9T^{2}
41 1+(3.74e4−2.16e4i)T+(2.37e9−4.11e9i)T2 1 + (3.74e4 - 2.16e4i)T + (2.37e9 - 4.11e9i)T^{2}
43 1+(−1.92e4+3.33e4i)T+(−3.16e9−5.47e9i)T2 1 + (-1.92e4 + 3.33e4i)T + (-3.16e9 - 5.47e9i)T^{2}
47 1+(1.43e5+8.30e4i)T+(5.38e9+9.33e9i)T2 1 + (1.43e5 + 8.30e4i)T + (5.38e9 + 9.33e9i)T^{2}
53 1+5.47e4iT−2.21e10T2 1 + 5.47e4iT - 2.21e10T^{2}
59 1+(1.41e4−8.14e3i)T+(2.10e10−3.65e10i)T2 1 + (1.41e4 - 8.14e3i)T + (2.10e10 - 3.65e10i)T^{2}
61 1+(−2.94e4+5.09e4i)T+(−2.57e10−4.46e10i)T2 1 + (-2.94e4 + 5.09e4i)T + (-2.57e10 - 4.46e10i)T^{2}
67 1+(1.47e5+2.56e5i)T+(−4.52e10+7.83e10i)T2 1 + (1.47e5 + 2.56e5i)T + (-4.52e10 + 7.83e10i)T^{2}
71 1−1.57e5iT−1.28e11T2 1 - 1.57e5iT - 1.28e11T^{2}
73 1−8.02e4T+1.51e11T2 1 - 8.02e4T + 1.51e11T^{2}
79 1+(−1.88e5+3.26e5i)T+(−1.21e11−2.10e11i)T2 1 + (-1.88e5 + 3.26e5i)T + (-1.21e11 - 2.10e11i)T^{2}
83 1+(−7.33e5−4.23e5i)T+(1.63e11+2.83e11i)T2 1 + (-7.33e5 - 4.23e5i)T + (1.63e11 + 2.83e11i)T^{2}
89 1−1.12e3iT−4.96e11T2 1 - 1.12e3iT - 4.96e11T^{2}
97 1+(−6.75e5+1.16e6i)T+(−4.16e11−7.21e11i)T2 1 + (-6.75e5 + 1.16e6i)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

−17.29139922706735846707004970736, −16.27074516525627931739407220420, −14.03467266089762857567130013832, −13.34860981476329313118283518343, −12.28653500730699933857477124867, −10.37389348947025042140834347808, −8.252838173412415976424399969281, −6.29300855829618507063050984240, −5.19218039057125298158701569428, −1.64997347428988153635377497744, 2.60353758399536868074866720700, 5.12952550875453119040663468064, 6.34625458706468386567629492009, 9.623049250838380313456543442972, 10.47026477636543706282173826006, 11.87379381084840535983161764829, 13.70063761268388598062152130672, 14.66190269839696483640443271229, 15.99699330801760358346650753135, 17.69116123045969451275203970486

Graph of the ZZ-function along the critical line