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Properties

Label 2-18e2-3.2-c8-0-1
Degree 22
Conductor 324324
Sign ii
Analytic cond. 131.990131.990
Root an. cond. 11.488711.4887
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11e3i·5-s + 285.·7-s + 2.21e4i·11-s − 3.84e4·13-s − 4.41e4i·17-s − 6.00e4·19-s + 2.89e5i·23-s − 8.55e5·25-s − 9.03e5i·29-s − 1.20e5·31-s + 3.18e5i·35-s − 2.67e6·37-s − 2.54e6i·41-s + 3.26e6·43-s + 9.69e5i·47-s + ⋯
L(s)  = 1  + 1.78i·5-s + 0.118·7-s + 1.51i·11-s − 1.34·13-s − 0.528i·17-s − 0.461·19-s + 1.03i·23-s − 2.19·25-s − 1.27i·29-s − 0.129·31-s + 0.212i·35-s − 1.42·37-s − 0.902i·41-s + 0.954·43-s + 0.198i·47-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=(i Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+4) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: ii
Analytic conductor: 131.990131.990
Root analytic conductor: 11.488711.4887
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ324(161,⋅)\chi_{324} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :4), i)(2,\ 324,\ (\ :4),\ i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.21255509520.2125550952
L(12)L(\frac12) ≈\approx 0.21255509520.2125550952
L(5)L(5) 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 1
3 1 1
good5 1−1.11e3iT−3.90e5T2 1 - 1.11e3iT - 3.90e5T^{2}
7 1−285.T+5.76e6T2 1 - 285.T + 5.76e6T^{2}
11 1−2.21e4iT−2.14e8T2 1 - 2.21e4iT - 2.14e8T^{2}
13 1+3.84e4T+8.15e8T2 1 + 3.84e4T + 8.15e8T^{2}
17 1+4.41e4iT−6.97e9T2 1 + 4.41e4iT - 6.97e9T^{2}
19 1+6.00e4T+1.69e10T2 1 + 6.00e4T + 1.69e10T^{2}
23 1−2.89e5iT−7.83e10T2 1 - 2.89e5iT - 7.83e10T^{2}
29 1+9.03e5iT−5.00e11T2 1 + 9.03e5iT - 5.00e11T^{2}
31 1+1.20e5T+8.52e11T2 1 + 1.20e5T + 8.52e11T^{2}
37 1+2.67e6T+3.51e12T2 1 + 2.67e6T + 3.51e12T^{2}
41 1+2.54e6iT−7.98e12T2 1 + 2.54e6iT - 7.98e12T^{2}
43 1−3.26e6T+1.16e13T2 1 - 3.26e6T + 1.16e13T^{2}
47 1−9.69e5iT−2.38e13T2 1 - 9.69e5iT - 2.38e13T^{2}
53 1−1.46e7iT−6.22e13T2 1 - 1.46e7iT - 6.22e13T^{2}
59 1−2.83e6iT−1.46e14T2 1 - 2.83e6iT - 1.46e14T^{2}
61 1+1.03e7T+1.91e14T2 1 + 1.03e7T + 1.91e14T^{2}
67 1−3.39e7T+4.06e14T2 1 - 3.39e7T + 4.06e14T^{2}
71 1−7.67e6iT−6.45e14T2 1 - 7.67e6iT - 6.45e14T^{2}
73 1−1.26e7T+8.06e14T2 1 - 1.26e7T + 8.06e14T^{2}
79 1−6.59e7T+1.51e15T2 1 - 6.59e7T + 1.51e15T^{2}
83 1+1.75e6iT−2.25e15T2 1 + 1.75e6iT - 2.25e15T^{2}
89 1−8.38e6iT−3.93e15T2 1 - 8.38e6iT - 3.93e15T^{2}
97 1−1.15e8T+7.83e15T2 1 - 1.15e8T + 7.83e15T^{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

−10.80406303242041297060349448305, −10.02614115073108065386260575472, −9.415221796809190324136199734053, −7.57884219111941284748611381798, −7.31123179622319059132133934433, −6.35136842824487178579287300397, −5.04434681147497821085779710720, −3.87429726202467499767408931418, −2.63175520755759583593173342304, −2.00275166007086312073887042956, 0.04940228754760040067426460507, 0.852229231062970287571024773968, 2.01341124605180104520993692086, 3.52818771670759771946365767942, 4.76131032959412641436553422263, 5.33100224682776416662148503002, 6.51352121123322101689262590407, 7.987256059256749100944379703034, 8.577549479876785577238154223877, 9.300949663142568383877316885161

Graph of the ZZ-function along the critical line