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Properties

Label 2-92-23.12-c5-0-3
Degree 22
Conductor 9292
Sign 0.999−0.0211i0.999 - 0.0211i
Analytic cond. 14.755314.7553
Root an. cond. 3.841263.84126
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.205 − 1.42i)3-s + (−104. + 30.5i)5-s + (−109. − 126. i)7-s + (231. + 67.8i)9-s + (468. + 300. i)11-s + (640. − 738. i)13-s + (22.2 + 155. i)15-s + (762. + 1.66e3i)17-s + (−16.7 + 36.6i)19-s + (−202. + 130. i)21-s + (−2.21e3 − 1.23e3i)23-s + (7.28e3 − 4.68e3i)25-s + (290. − 635. i)27-s + (568. + 1.24e3i)29-s + (519. + 3.61e3i)31-s + ⋯
L(s)  = 1  + (0.0131 − 0.0916i)3-s + (−1.86 + 0.547i)5-s + (−0.843 − 0.972i)7-s + (0.951 + 0.279i)9-s + (1.16 + 0.749i)11-s + (1.05 − 1.21i)13-s + (0.0255 + 0.177i)15-s + (0.639 + 1.40i)17-s + (−0.0106 + 0.0232i)19-s + (−0.100 + 0.0644i)21-s + (−0.872 − 0.487i)23-s + (2.33 − 1.49i)25-s + (0.0765 − 0.167i)27-s + (0.125 + 0.274i)29-s + (0.0971 + 0.675i)31-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((0.999−0.0211i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+5/2) L(s)=((0.999−0.0211i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: 0.999−0.0211i0.999 - 0.0211i
Analytic conductor: 14.755314.7553
Root analytic conductor: 3.841263.84126
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ92(81,⋅)\chi_{92} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :5/2), 0.999−0.0211i)(2,\ 92,\ (\ :5/2),\ 0.999 - 0.0211i)

Particular Values

L(3)L(3) ≈\approx 1.30636+0.0138419i1.30636 + 0.0138419i
L(12)L(\frac12) ≈\approx 1.30636+0.0138419i1.30636 + 0.0138419i
L(72)L(\frac{7}{2}) 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
23 1+(2.21e3+1.23e3i)T 1 + (2.21e3 + 1.23e3i)T
good3 1+(−0.205+1.42i)T+(−233.−68.4i)T2 1 + (-0.205 + 1.42i)T + (-233. - 68.4i)T^{2}
5 1+(104.−30.5i)T+(2.62e3−1.68e3i)T2 1 + (104. - 30.5i)T + (2.62e3 - 1.68e3i)T^{2}
7 1+(109.+126.i)T+(−2.39e3+1.66e4i)T2 1 + (109. + 126. i)T + (-2.39e3 + 1.66e4i)T^{2}
11 1+(−468.−300.i)T+(6.69e4+1.46e5i)T2 1 + (-468. - 300. i)T + (6.69e4 + 1.46e5i)T^{2}
13 1+(−640.+738.i)T+(−5.28e4−3.67e5i)T2 1 + (-640. + 738. i)T + (-5.28e4 - 3.67e5i)T^{2}
17 1+(−762.−1.66e3i)T+(−9.29e5+1.07e6i)T2 1 + (-762. - 1.66e3i)T + (-9.29e5 + 1.07e6i)T^{2}
19 1+(16.7−36.6i)T+(−1.62e6−1.87e6i)T2 1 + (16.7 - 36.6i)T + (-1.62e6 - 1.87e6i)T^{2}
29 1+(−568.−1.24e3i)T+(−1.34e7+1.55e7i)T2 1 + (-568. - 1.24e3i)T + (-1.34e7 + 1.55e7i)T^{2}
31 1+(−519.−3.61e3i)T+(−2.74e7+8.06e6i)T2 1 + (-519. - 3.61e3i)T + (-2.74e7 + 8.06e6i)T^{2}
37 1+(−2.05e3−602.i)T+(5.83e7+3.74e7i)T2 1 + (-2.05e3 - 602. i)T + (5.83e7 + 3.74e7i)T^{2}
41 1+(−5.79e3+1.70e3i)T+(9.74e7−6.26e7i)T2 1 + (-5.79e3 + 1.70e3i)T + (9.74e7 - 6.26e7i)T^{2}
43 1+(−1.76e3+1.22e4i)T+(−1.41e8−4.14e7i)T2 1 + (-1.76e3 + 1.22e4i)T + (-1.41e8 - 4.14e7i)T^{2}
47 1−1.77e4T+2.29e8T2 1 - 1.77e4T + 2.29e8T^{2}
53 1+(3.18e3+3.67e3i)T+(−5.95e7+4.13e8i)T2 1 + (3.18e3 + 3.67e3i)T + (-5.95e7 + 4.13e8i)T^{2}
59 1+(−7.04e3+8.13e3i)T+(−1.01e8−7.07e8i)T2 1 + (-7.04e3 + 8.13e3i)T + (-1.01e8 - 7.07e8i)T^{2}
61 1+(−2.15e3−1.49e4i)T+(−8.10e8+2.37e8i)T2 1 + (-2.15e3 - 1.49e4i)T + (-8.10e8 + 2.37e8i)T^{2}
67 1+(−3.57e4+2.29e4i)T+(5.60e8−1.22e9i)T2 1 + (-3.57e4 + 2.29e4i)T + (5.60e8 - 1.22e9i)T^{2}
71 1+(−3.23e4+2.08e4i)T+(7.49e8−1.64e9i)T2 1 + (-3.23e4 + 2.08e4i)T + (7.49e8 - 1.64e9i)T^{2}
73 1+(1.71e4−3.76e4i)T+(−1.35e9−1.56e9i)T2 1 + (1.71e4 - 3.76e4i)T + (-1.35e9 - 1.56e9i)T^{2}
79 1+(4.13e4−4.76e4i)T+(−4.37e8−3.04e9i)T2 1 + (4.13e4 - 4.76e4i)T + (-4.37e8 - 3.04e9i)T^{2}
83 1+(−1.02e4−3.01e3i)T+(3.31e9+2.12e9i)T2 1 + (-1.02e4 - 3.01e3i)T + (3.31e9 + 2.12e9i)T^{2}
89 1+(1.64e4−1.14e5i)T+(−5.35e9−1.57e9i)T2 1 + (1.64e4 - 1.14e5i)T + (-5.35e9 - 1.57e9i)T^{2}
97 1+(−8.11e4+2.38e4i)T+(7.22e9−4.64e9i)T2 1 + (-8.11e4 + 2.38e4i)T + (7.22e9 - 4.64e9i)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

−12.80653151135812007547894008752, −12.22246963853905420931736529333, −10.78732429873552411217998569246, −10.19421235578724598753005232224, −8.323238660923036938962787014239, −7.39778411895354508161696732525, −6.53419789821950847852823500612, −4.04826550191508799573772793825, −3.65848405475102204938017143331, −0.878652919611415370364722331655, 0.862620232942329620137426373118, 3.48039367669819132561797405899, 4.31504247313583446473896950418, 6.23013390466170782622118423569, 7.45694471681466119760695663434, 8.792565950376566812556391386060, 9.451968258929096507075813062261, 11.53473207542838542240167763263, 11.77726299475951208071687173774, 12.83119663184158574702506166400

Graph of the ZZ-function along the critical line