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Properties

Label 2-675-27.16-c1-0-6
Degree 22
Conductor 675675
Sign 0.823−0.567i0.823 - 0.567i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.55 − 1.30i)2-s + (−1.45 + 0.933i)3-s + (0.364 + 2.06i)4-s + (3.47 + 0.449i)6-s + (0.0652 − 0.370i)7-s + (0.101 − 0.176i)8-s + (1.25 − 2.72i)9-s + (0.272 − 0.0993i)11-s + (−2.46 − 2.67i)12-s + (−0.677 + 0.568i)13-s + (−0.583 + 0.489i)14-s + (3.56 − 1.29i)16-s + (−2.32 − 4.02i)17-s + (−5.49 + 2.59i)18-s + (−1.75 + 3.03i)19-s + ⋯
L(s)  = 1  + (−1.09 − 0.920i)2-s + (−0.842 + 0.539i)3-s + (0.182 + 1.03i)4-s + (1.42 + 0.183i)6-s + (0.0246 − 0.139i)7-s + (0.0359 − 0.0622i)8-s + (0.418 − 0.908i)9-s + (0.0823 − 0.0299i)11-s + (−0.711 − 0.772i)12-s + (−0.187 + 0.157i)13-s + (−0.155 + 0.130i)14-s + (0.890 − 0.323i)16-s + (−0.563 − 0.976i)17-s + (−1.29 + 0.611i)18-s + (−0.401 + 0.695i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.823−0.567i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.823−0.567i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.823−0.567i0.823 - 0.567i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(151,⋅)\chi_{675} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.823−0.567i)(2,\ 675,\ (\ :1/2),\ 0.823 - 0.567i)

Particular Values

L(1)L(1) ≈\approx 0.394311+0.122648i0.394311 + 0.122648i
L(12)L(\frac12) ≈\approx 0.394311+0.122648i0.394311 + 0.122648i
L(32)L(\frac{3}{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)
bad3 1+(1.45−0.933i)T 1 + (1.45 - 0.933i)T
5 1 1
good2 1+(1.55+1.30i)T+(0.347+1.96i)T2 1 + (1.55 + 1.30i)T + (0.347 + 1.96i)T^{2}
7 1+(−0.0652+0.370i)T+(−6.57−2.39i)T2 1 + (-0.0652 + 0.370i)T + (-6.57 - 2.39i)T^{2}
11 1+(−0.272+0.0993i)T+(8.42−7.07i)T2 1 + (-0.272 + 0.0993i)T + (8.42 - 7.07i)T^{2}
13 1+(0.677−0.568i)T+(2.25−12.8i)T2 1 + (0.677 - 0.568i)T + (2.25 - 12.8i)T^{2}
17 1+(2.32+4.02i)T+(−8.5+14.7i)T2 1 + (2.32 + 4.02i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.75−3.03i)T+(−9.5−16.4i)T2 1 + (1.75 - 3.03i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.0948+0.537i)T+(−21.6+7.86i)T2 1 + (0.0948 + 0.537i)T + (-21.6 + 7.86i)T^{2}
29 1+(4.65+3.90i)T+(5.03+28.5i)T2 1 + (4.65 + 3.90i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.953−5.40i)T+(−29.1+10.6i)T2 1 + (-0.953 - 5.40i)T + (-29.1 + 10.6i)T^{2}
37 1+(−5.47−9.48i)T+(−18.5+32.0i)T2 1 + (-5.47 - 9.48i)T + (-18.5 + 32.0i)T^{2}
41 1+(7.28−6.11i)T+(7.11−40.3i)T2 1 + (7.28 - 6.11i)T + (7.11 - 40.3i)T^{2}
43 1+(−9.54+3.47i)T+(32.9−27.6i)T2 1 + (-9.54 + 3.47i)T + (32.9 - 27.6i)T^{2}
47 1+(1.09−6.23i)T+(−44.1−16.0i)T2 1 + (1.09 - 6.23i)T + (-44.1 - 16.0i)T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1+(1.18+0.432i)T+(45.1+37.9i)T2 1 + (1.18 + 0.432i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.499+2.83i)T+(−57.3−20.8i)T2 1 + (-0.499 + 2.83i)T + (-57.3 - 20.8i)T^{2}
67 1+(6.78−5.69i)T+(11.6−65.9i)T2 1 + (6.78 - 5.69i)T + (11.6 - 65.9i)T^{2}
71 1+(−5.36−9.29i)T+(−35.5+61.4i)T2 1 + (-5.36 - 9.29i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.389+0.674i)T+(−36.5−63.2i)T2 1 + (-0.389 + 0.674i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5.29−4.44i)T+(13.7+77.7i)T2 1 + (-5.29 - 4.44i)T + (13.7 + 77.7i)T^{2}
83 1+(−6.88−5.77i)T+(14.4+81.7i)T2 1 + (-6.88 - 5.77i)T + (14.4 + 81.7i)T^{2}
89 1+(3.11−5.38i)T+(−44.5−77.0i)T2 1 + (3.11 - 5.38i)T + (-44.5 - 77.0i)T^{2}
97 1+(13.1−4.79i)T+(74.3−62.3i)T2 1 + (13.1 - 4.79i)T + (74.3 - 62.3i)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

−10.52234951230043697248086287750, −9.840348990206574738996116196091, −9.226106606382658666032147561194, −8.307348021549343635640449966776, −7.18903475470814063319667309335, −6.11605379437881612844838642962, −5.03648770069329825986481605031, −3.93175024733546852305157035703, −2.58850330367260435836384117536, −1.06478601189250422284587249680, 0.43064993584842492855702380591, 2.04657489386767476428695165647, 4.07096316557271757081150308469, 5.47857956746317986793217944165, 6.15437056767203770731245203560, 7.05482091456594307084002682869, 7.62556774870938086356923744265, 8.607069669557467518372935749060, 9.316865280991189701768725136864, 10.43037531847494923503517003438

Graph of the ZZ-function along the critical line