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Properties

Label 2-99-11.3-c3-0-4
Degree 22
Conductor 9999
Sign 0.621−0.783i0.621 - 0.783i
Analytic cond. 5.841185.84118
Root an. cond. 2.416852.41685
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 − 1.01i)2-s + (−1.54 + 4.76i)4-s + (3.78 + 2.75i)5-s + (−2.91 + 8.97i)7-s + (6.95 + 21.3i)8-s + 8.09·10-s + (20.8 + 29.9i)11-s + (−4.36 + 3.16i)13-s + (5.04 + 15.5i)14-s + (−0.939 − 0.682i)16-s + (68.6 + 49.8i)17-s + (−15.7 − 48.4i)19-s + (−18.9 + 13.7i)20-s + (59.6 + 20.5i)22-s + 52.9·23-s + ⋯
L(s)  = 1  + (0.494 − 0.359i)2-s + (−0.193 + 0.595i)4-s + (0.338 + 0.246i)5-s + (−0.157 + 0.484i)7-s + (0.307 + 0.945i)8-s + 0.256·10-s + (0.572 + 0.819i)11-s + (−0.0930 + 0.0676i)13-s + (0.0962 + 0.296i)14-s + (−0.0146 − 0.0106i)16-s + (0.979 + 0.711i)17-s + (−0.189 − 0.584i)19-s + (−0.212 + 0.154i)20-s + (0.577 + 0.199i)22-s + 0.480·23-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((0.621−0.783i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+3/2) L(s)=((0.621−0.783i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 0.621−0.783i0.621 - 0.783i
Analytic conductor: 5.841185.84118
Root analytic conductor: 2.416852.41685
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ99(91,⋅)\chi_{99} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :3/2), 0.621−0.783i)(2,\ 99,\ (\ :3/2),\ 0.621 - 0.783i)

Particular Values

L(2)L(2) ≈\approx 1.74802+0.844803i1.74802 + 0.844803i
L(12)L(\frac12) ≈\approx 1.74802+0.844803i1.74802 + 0.844803i
L(52)L(\frac{5}{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
11 1+(−20.8−29.9i)T 1 + (-20.8 - 29.9i)T
good2 1+(−1.39+1.01i)T+(2.47−7.60i)T2 1 + (-1.39 + 1.01i)T + (2.47 - 7.60i)T^{2}
5 1+(−3.78−2.75i)T+(38.6+118.i)T2 1 + (-3.78 - 2.75i)T + (38.6 + 118. i)T^{2}
7 1+(2.91−8.97i)T+(−277.−201.i)T2 1 + (2.91 - 8.97i)T + (-277. - 201. i)T^{2}
13 1+(4.36−3.16i)T+(678.−2.08e3i)T2 1 + (4.36 - 3.16i)T + (678. - 2.08e3i)T^{2}
17 1+(−68.6−49.8i)T+(1.51e3+4.67e3i)T2 1 + (-68.6 - 49.8i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(15.7+48.4i)T+(−5.54e3+4.03e3i)T2 1 + (15.7 + 48.4i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−52.9T+1.21e4T2 1 - 52.9T + 1.21e4T^{2}
29 1+(−35.2+108.i)T+(−1.97e4−1.43e4i)T2 1 + (-35.2 + 108. i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(67.7−49.2i)T+(9.20e3−2.83e4i)T2 1 + (67.7 - 49.2i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(47.0−144.i)T+(−4.09e4−2.97e4i)T2 1 + (47.0 - 144. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(56.9+175.i)T+(−5.57e4+4.05e4i)T2 1 + (56.9 + 175. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+46.4T+7.95e4T2 1 + 46.4T + 7.95e4T^{2}
47 1+(120.+370.i)T+(−8.39e4+6.10e4i)T2 1 + (120. + 370. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−462.+336.i)T+(4.60e4−1.41e5i)T2 1 + (-462. + 336. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(24.4−75.2i)T+(−1.66e5−1.20e5i)T2 1 + (24.4 - 75.2i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(584.+424.i)T+(7.01e4+2.15e5i)T2 1 + (584. + 424. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−608.T+3.00e5T2 1 - 608.T + 3.00e5T^{2}
71 1+(−740.−538.i)T+(1.10e5+3.40e5i)T2 1 + (-740. - 538. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−186.+573.i)T+(−3.14e5−2.28e5i)T2 1 + (-186. + 573. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−298.+216.i)T+(1.52e5−4.68e5i)T2 1 + (-298. + 216. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−797.−579.i)T+(1.76e5+5.43e5i)T2 1 + (-797. - 579. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−1.15e3T+7.04e5T2 1 - 1.15e3T + 7.04e5T^{2}
97 1+(478.−347.i)T+(2.82e5−8.68e5i)T2 1 + (478. - 347. i)T + (2.82e5 - 8.68e5i)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

−13.45028158035892843837904566938, −12.41702276478503258718022608606, −11.81907516287808221030057547472, −10.43489062497329870943098369671, −9.234975674565767722484226911202, −8.053650146343555596284019139384, −6.66525153679442087417661534095, −5.14325897101030735619610388746, −3.74787565407092752980227896322, −2.25502104823458872780668015815, 1.06963482785208819969545338532, 3.63739156235709203020585204622, 5.14285591681447946331404397706, 6.13359944450733873357708996610, 7.38807581628054247742677887034, 9.030866078188901265432637944387, 9.963521991545711654355383055948, 11.06367878001109718675575423498, 12.46926645368263375543629797042, 13.55900447301162316614800623418

Graph of the ZZ-function along the critical line