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Properties

Label 2-18-9.2-c10-0-2
Degree 22
Conductor 1818
Sign 0.0370−0.999i0.0370 - 0.999i
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 + (16.6 + 242. i)3-s + (255. + 443. i)4-s + (2.72e3 − 1.57e3i)5-s + (2.41e3 − 4.93e3i)6-s + (−1.76e3 + 3.06e3i)7-s − 1.15e4i·8-s + (−5.84e4 + 8.09e3i)9-s − 7.11e4·10-s + (1.01e5 + 5.88e4i)11-s + (−1.03e5 + 6.94e4i)12-s + (1.40e5 + 2.42e5i)13-s + (6.93e4 − 4.00e4i)14-s + (4.26e5 + 6.33e5i)15-s + (−1.31e5 + 2.27e5i)16-s + 2.62e6i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.0687 + 0.997i)3-s + (0.249 + 0.433i)4-s + (0.871 − 0.503i)5-s + (0.310 − 0.635i)6-s + (−0.105 + 0.182i)7-s − 0.353i·8-s + (−0.990 + 0.137i)9-s − 0.711·10-s + (0.632 + 0.365i)11-s + (−0.414 + 0.279i)12-s + (0.377 + 0.653i)13-s + (0.128 − 0.0744i)14-s + (0.561 + 0.834i)15-s + (−0.125 + 0.216i)16-s + 1.84i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((0.0370−0.999i) Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0370 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+5) L(s)=((0.0370−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0370 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: 0.0370−0.999i0.0370 - 0.999i
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), 0.0370−0.999i)(2,\ 18,\ (\ :5),\ 0.0370 - 0.999i)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 0.954562+0.919869i0.954562 + 0.919869i
L(12)L(\frac12) ≈\approx 0.954562+0.919869i0.954562 + 0.919869i
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+(−16.6−242.i)T 1 + (-16.6 - 242. i)T
good5 1+(−2.72e3+1.57e3i)T+(4.88e6−8.45e6i)T2 1 + (-2.72e3 + 1.57e3i)T + (4.88e6 - 8.45e6i)T^{2}
7 1+(1.76e3−3.06e3i)T+(−1.41e8−2.44e8i)T2 1 + (1.76e3 - 3.06e3i)T + (-1.41e8 - 2.44e8i)T^{2}
11 1+(−1.01e5−5.88e4i)T+(1.29e10+2.24e10i)T2 1 + (-1.01e5 - 5.88e4i)T + (1.29e10 + 2.24e10i)T^{2}
13 1+(−1.40e5−2.42e5i)T+(−6.89e10+1.19e11i)T2 1 + (-1.40e5 - 2.42e5i)T + (-6.89e10 + 1.19e11i)T^{2}
17 1−2.62e6iT−2.01e12T2 1 - 2.62e6iT - 2.01e12T^{2}
19 1+1.16e6T+6.13e12T2 1 + 1.16e6T + 6.13e12T^{2}
23 1+(6.93e6−4.00e6i)T+(2.07e13−3.58e13i)T2 1 + (6.93e6 - 4.00e6i)T + (2.07e13 - 3.58e13i)T^{2}
29 1+(−2.58e7−1.49e7i)T+(2.10e14+3.64e14i)T2 1 + (-2.58e7 - 1.49e7i)T + (2.10e14 + 3.64e14i)T^{2}
31 1+(9.71e6+1.68e7i)T+(−4.09e14+7.09e14i)T2 1 + (9.71e6 + 1.68e7i)T + (-4.09e14 + 7.09e14i)T^{2}
37 1+2.58e7T+4.80e15T2 1 + 2.58e7T + 4.80e15T^{2}
41 1+(3.60e7−2.08e7i)T+(6.71e15−1.16e16i)T2 1 + (3.60e7 - 2.08e7i)T + (6.71e15 - 1.16e16i)T^{2}
43 1+(−9.70e7+1.68e8i)T+(−1.08e16−1.87e16i)T2 1 + (-9.70e7 + 1.68e8i)T + (-1.08e16 - 1.87e16i)T^{2}
47 1+(−2.63e8−1.51e8i)T+(2.62e16+4.55e16i)T2 1 + (-2.63e8 - 1.51e8i)T + (2.62e16 + 4.55e16i)T^{2}
53 1−2.93e8iT−1.74e17T2 1 - 2.93e8iT - 1.74e17T^{2}
59 1+(−4.61e8+2.66e8i)T+(2.55e17−4.42e17i)T2 1 + (-4.61e8 + 2.66e8i)T + (2.55e17 - 4.42e17i)T^{2}
61 1+(−5.88e8+1.01e9i)T+(−3.56e17−6.17e17i)T2 1 + (-5.88e8 + 1.01e9i)T + (-3.56e17 - 6.17e17i)T^{2}
67 1+(9.91e8+1.71e9i)T+(−9.11e17+1.57e18i)T2 1 + (9.91e8 + 1.71e9i)T + (-9.11e17 + 1.57e18i)T^{2}
71 1+1.40e9iT−3.25e18T2 1 + 1.40e9iT - 3.25e18T^{2}
73 1+6.62e7T+4.29e18T2 1 + 6.62e7T + 4.29e18T^{2}
79 1+(2.94e9−5.10e9i)T+(−4.73e18−8.19e18i)T2 1 + (2.94e9 - 5.10e9i)T + (-4.73e18 - 8.19e18i)T^{2}
83 1+(−2.24e9−1.29e9i)T+(7.75e18+1.34e19i)T2 1 + (-2.24e9 - 1.29e9i)T + (7.75e18 + 1.34e19i)T^{2}
89 1+8.23e9iT−3.11e19T2 1 + 8.23e9iT - 3.11e19T^{2}
97 1+(6.95e9−1.20e10i)T+(−3.68e19−6.38e19i)T2 1 + (6.95e9 - 1.20e10i)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

−16.86235015692985114212574641458, −15.51491895844238825380269160146, −14.01867532573532465416364750890, −12.32912265472672438475422371242, −10.71227080164419263965403979208, −9.563308408843295656298843247272, −8.586957907161055876073032619010, −6.01361702542906465648343580157, −4.00959150960637542568399952362, −1.80547518043024740498930227835, 0.73058395315191136907555815427, 2.49873817544010628789556060786, 5.92849115647731593224658871335, 7.02754528840577214198758964916, 8.616345229672201391400761136210, 10.18551708470563750436416741073, 11.79204850084749136116601318789, 13.57437599697071137585020666442, 14.37420545099211937379605280278, 16.22729565618462854158235594064

Graph of the ZZ-function along the critical line