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Properties

Label 2-18-9.2-c10-0-8
Degree 22
Conductor 1818
Sign −0.989+0.144i-0.989 + 0.144i
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 + (−196. − 142. i)3-s + (255. + 443. i)4-s + (3.09e3 − 1.78e3i)5-s + (2.25e3 + 5.01e3i)6-s + (1.26e4 − 2.18e4i)7-s − 1.15e4i·8-s + (1.85e4 + 5.60e4i)9-s − 8.07e4·10-s + (−1.71e5 − 9.91e4i)11-s + (1.26e4 − 1.23e5i)12-s + (1.41e5 + 2.45e5i)13-s + (−4.93e5 + 2.85e5i)14-s + (−8.63e5 − 8.82e4i)15-s + (−1.31e5 + 2.27e5i)16-s − 1.47e6i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.810 − 0.585i)3-s + (0.249 + 0.433i)4-s + (0.989 − 0.571i)5-s + (0.289 + 0.645i)6-s + (0.749 − 1.29i)7-s − 0.353i·8-s + (0.314 + 0.949i)9-s − 0.807·10-s + (−1.06 − 0.615i)11-s + (0.0508 − 0.497i)12-s + (0.382 + 0.662i)13-s + (−0.918 + 0.530i)14-s + (−1.13 − 0.116i)15-s + (−0.125 + 0.216i)16-s − 1.03i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.989+0.144i) Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+5) L(s)=((−0.989+0.144i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.989+0.144i-0.989 + 0.144i
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), −0.989+0.144i)(2,\ 18,\ (\ :5),\ -0.989 + 0.144i)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 0.0622268−0.855729i0.0622268 - 0.855729i
L(12)L(\frac12) ≈\approx 0.0622268−0.855729i0.0622268 - 0.855729i
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+(196.+142.i)T 1 + (196. + 142. i)T
good5 1+(−3.09e3+1.78e3i)T+(4.88e6−8.45e6i)T2 1 + (-3.09e3 + 1.78e3i)T + (4.88e6 - 8.45e6i)T^{2}
7 1+(−1.26e4+2.18e4i)T+(−1.41e8−2.44e8i)T2 1 + (-1.26e4 + 2.18e4i)T + (-1.41e8 - 2.44e8i)T^{2}
11 1+(1.71e5+9.91e4i)T+(1.29e10+2.24e10i)T2 1 + (1.71e5 + 9.91e4i)T + (1.29e10 + 2.24e10i)T^{2}
13 1+(−1.41e5−2.45e5i)T+(−6.89e10+1.19e11i)T2 1 + (-1.41e5 - 2.45e5i)T + (-6.89e10 + 1.19e11i)T^{2}
17 1+1.47e6iT−2.01e12T2 1 + 1.47e6iT - 2.01e12T^{2}
19 1+4.67e6T+6.13e12T2 1 + 4.67e6T + 6.13e12T^{2}
23 1+(−4.51e6+2.60e6i)T+(2.07e13−3.58e13i)T2 1 + (-4.51e6 + 2.60e6i)T + (2.07e13 - 3.58e13i)T^{2}
29 1+(−1.64e6−9.47e5i)T+(2.10e14+3.64e14i)T2 1 + (-1.64e6 - 9.47e5i)T + (2.10e14 + 3.64e14i)T^{2}
31 1+(−9.25e6−1.60e7i)T+(−4.09e14+7.09e14i)T2 1 + (-9.25e6 - 1.60e7i)T + (-4.09e14 + 7.09e14i)T^{2}
37 1+9.06e7T+4.80e15T2 1 + 9.06e7T + 4.80e15T^{2}
41 1+(4.27e6−2.46e6i)T+(6.71e15−1.16e16i)T2 1 + (4.27e6 - 2.46e6i)T + (6.71e15 - 1.16e16i)T^{2}
43 1+(3.66e7−6.34e7i)T+(−1.08e16−1.87e16i)T2 1 + (3.66e7 - 6.34e7i)T + (-1.08e16 - 1.87e16i)T^{2}
47 1+(2.91e8+1.68e8i)T+(2.62e16+4.55e16i)T2 1 + (2.91e8 + 1.68e8i)T + (2.62e16 + 4.55e16i)T^{2}
53 1+8.02e7iT−1.74e17T2 1 + 8.02e7iT - 1.74e17T^{2}
59 1+(−7.95e8+4.59e8i)T+(2.55e17−4.42e17i)T2 1 + (-7.95e8 + 4.59e8i)T + (2.55e17 - 4.42e17i)T^{2}
61 1+(−3.70e6+6.40e6i)T+(−3.56e17−6.17e17i)T2 1 + (-3.70e6 + 6.40e6i)T + (-3.56e17 - 6.17e17i)T^{2}
67 1+(−9.55e8−1.65e9i)T+(−9.11e17+1.57e18i)T2 1 + (-9.55e8 - 1.65e9i)T + (-9.11e17 + 1.57e18i)T^{2}
71 1+2.72e9iT−3.25e18T2 1 + 2.72e9iT - 3.25e18T^{2}
73 1−2.42e9T+4.29e18T2 1 - 2.42e9T + 4.29e18T^{2}
79 1+(−7.42e8+1.28e9i)T+(−4.73e18−8.19e18i)T2 1 + (-7.42e8 + 1.28e9i)T + (-4.73e18 - 8.19e18i)T^{2}
83 1+(5.66e8+3.27e8i)T+(7.75e18+1.34e19i)T2 1 + (5.66e8 + 3.27e8i)T + (7.75e18 + 1.34e19i)T^{2}
89 1−1.33e9iT−3.11e19T2 1 - 1.33e9iT - 3.11e19T^{2}
97 1+(−2.78e9+4.81e9i)T+(−3.68e19−6.38e19i)T2 1 + (-2.78e9 + 4.81e9i)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

−16.38435884787543085059627031623, −13.80326348672491228654742529742, −12.96150198020109200605410607191, −11.20585370238956063668057998433, −10.32732223722827823179007941001, −8.385844726350783403331963201947, −6.79349976835674107933899889980, −4.92803652081556773541283554004, −1.81821332367307356582676679757, −0.50560951626531010943824420861, 2.09996638065391272464285291617, 5.24879052237759808424602116615, 6.28226063445411451552158013387, 8.497882665233609484487357319819, 10.08156948335707824794903659955, 10.95942346561221919009470910710, 12.71169475263731220418146275648, 14.92161352680884564319368457042, 15.45226074568206552695716650081, 17.27273312174851836828526504659

Graph of the ZZ-function along the critical line