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Properties

Label 2-18-3.2-c12-0-0
Degree 22
Conductor 1818
Sign −0.816+0.577i-0.816 + 0.577i
Analytic cond. 16.451816.4518
Root an. cond. 4.056094.05609
Motivic weight 1212
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2i·2-s − 2.04e3·4-s + 2.05e4i·5-s + 3.35e4·7-s − 9.26e4i·8-s − 9.32e5·10-s + 1.71e6i·11-s − 5.19e6·13-s + 1.51e6i·14-s + 4.19e6·16-s − 3.13e7i·17-s − 6.60e7·19-s − 4.21e7i·20-s − 7.75e7·22-s − 1.72e8i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.31i·5-s + 0.285·7-s − 0.353i·8-s − 0.932·10-s + 0.967i·11-s − 1.07·13-s + 0.201i·14-s + 0.250·16-s − 1.29i·17-s − 1.40·19-s − 0.659i·20-s − 0.684·22-s − 1.16i·23-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.816+0.577i) Λ‾(13−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(13-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+6) L(s)=((−0.816+0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.816+0.577i-0.816 + 0.577i
Analytic conductor: 16.451816.4518
Root analytic conductor: 4.056094.05609
Motivic weight: 1212
Rational: no
Arithmetic: yes
Character: χ18(17,⋅)\chi_{18} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :6), −0.816+0.577i)(2,\ 18,\ (\ :6),\ -0.816 + 0.577i)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 0.220889−0.694977i0.220889 - 0.694977i
L(12)L(\frac12) ≈\approx 0.220889−0.694977i0.220889 - 0.694977i
L(7)L(7) 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 1−45.2iT 1 - 45.2iT
3 1 1
good5 1−2.05e4iT−2.44e8T2 1 - 2.05e4iT - 2.44e8T^{2}
7 1−3.35e4T+1.38e10T2 1 - 3.35e4T + 1.38e10T^{2}
11 1−1.71e6iT−3.13e12T2 1 - 1.71e6iT - 3.13e12T^{2}
13 1+5.19e6T+2.32e13T2 1 + 5.19e6T + 2.32e13T^{2}
17 1+3.13e7iT−5.82e14T2 1 + 3.13e7iT - 5.82e14T^{2}
19 1+6.60e7T+2.21e15T2 1 + 6.60e7T + 2.21e15T^{2}
23 1+1.72e8iT−2.19e16T2 1 + 1.72e8iT - 2.19e16T^{2}
29 1−4.04e8iT−3.53e17T2 1 - 4.04e8iT - 3.53e17T^{2}
31 1+1.73e9T+7.87e17T2 1 + 1.73e9T + 7.87e17T^{2}
37 1−4.29e9T+6.58e18T2 1 - 4.29e9T + 6.58e18T^{2}
41 1−8.42e9iT−2.25e19T2 1 - 8.42e9iT - 2.25e19T^{2}
43 1−2.71e9T+3.99e19T2 1 - 2.71e9T + 3.99e19T^{2}
47 1−8.00e9iT−1.16e20T2 1 - 8.00e9iT - 1.16e20T^{2}
53 1−1.16e10iT−4.91e20T2 1 - 1.16e10iT - 4.91e20T^{2}
59 1+1.07e10iT−1.77e21T2 1 + 1.07e10iT - 1.77e21T^{2}
61 1+3.74e10T+2.65e21T2 1 + 3.74e10T + 2.65e21T^{2}
67 1−7.46e10T+8.18e21T2 1 - 7.46e10T + 8.18e21T^{2}
71 1−7.21e10iT−1.64e22T2 1 - 7.21e10iT - 1.64e22T^{2}
73 1+7.22e10T+2.29e22T2 1 + 7.22e10T + 2.29e22T^{2}
79 1−3.17e11T+5.90e22T2 1 - 3.17e11T + 5.90e22T^{2}
83 1+1.22e11iT−1.06e23T2 1 + 1.22e11iT - 1.06e23T^{2}
89 1−2.36e11iT−2.46e23T2 1 - 2.36e11iT - 2.46e23T^{2}
97 1+1.34e11T+6.93e23T2 1 + 1.34e11T + 6.93e23T^{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

−16.51921311414961822109780103279, −14.79587223845900709831801292028, −14.58672182690319023663664124672, −12.68639789252526026843061598844, −10.94482426352989567945839697814, −9.562971217129862458655237462806, −7.58748045305022106002296336561, −6.60545183937517885039668823130, −4.65397331821090732381970650134, −2.53436428829489754739677558188, 0.27309252011008683145190181076, 1.83820940461222875089441113121, 4.04732245275144891177602329047, 5.52738222820608218839524591322, 8.143873876351027827017920751480, 9.279779321535803293214060340964, 10.92304159676229824796348545054, 12.35599652170272402860594893378, 13.23799105504029971697980062431, 14.84229774903433363192870780566

Graph of the ZZ-function along the critical line