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Properties

Label 2-99-11.3-c3-0-1
Degree 22
Conductor 9999
Sign −0.884+0.466i-0.884 + 0.466i
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  + (−3.61 + 2.62i)2-s + (3.70 − 11.4i)4-s + (4.69 + 3.40i)5-s + (−4.72 + 14.5i)7-s + (5.52 + 17.0i)8-s − 25.9·10-s + (7.07 + 35.7i)11-s + (−37.6 + 27.3i)13-s + (−21.1 − 65.0i)14-s + (12.9 + 9.40i)16-s + (−91.6 − 66.5i)17-s + (−34.2 − 105. i)19-s + (56.2 − 40.8i)20-s + (−119. − 110. i)22-s − 12.1·23-s + ⋯
L(s)  = 1  + (−1.27 + 0.929i)2-s + (0.463 − 1.42i)4-s + (0.419 + 0.304i)5-s + (−0.255 + 0.785i)7-s + (0.244 + 0.751i)8-s − 0.820·10-s + (0.194 + 0.980i)11-s + (−0.804 + 0.584i)13-s + (−0.403 − 1.24i)14-s + (0.202 + 0.146i)16-s + (−1.30 − 0.949i)17-s + (−0.413 − 1.27i)19-s + (0.629 − 0.457i)20-s + (−1.15 − 1.07i)22-s − 0.110·23-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((−0.884+0.466i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+3/2) L(s)=((−0.884+0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: −0.884+0.466i-0.884 + 0.466i
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), −0.884+0.466i)(2,\ 99,\ (\ :3/2),\ -0.884 + 0.466i)

Particular Values

L(2)L(2) ≈\approx 0.0844997−0.341101i0.0844997 - 0.341101i
L(12)L(\frac12) ≈\approx 0.0844997−0.341101i0.0844997 - 0.341101i
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+(−7.07−35.7i)T 1 + (-7.07 - 35.7i)T
good2 1+(3.61−2.62i)T+(2.47−7.60i)T2 1 + (3.61 - 2.62i)T + (2.47 - 7.60i)T^{2}
5 1+(−4.69−3.40i)T+(38.6+118.i)T2 1 + (-4.69 - 3.40i)T + (38.6 + 118. i)T^{2}
7 1+(4.72−14.5i)T+(−277.−201.i)T2 1 + (4.72 - 14.5i)T + (-277. - 201. i)T^{2}
13 1+(37.6−27.3i)T+(678.−2.08e3i)T2 1 + (37.6 - 27.3i)T + (678. - 2.08e3i)T^{2}
17 1+(91.6+66.5i)T+(1.51e3+4.67e3i)T2 1 + (91.6 + 66.5i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(34.2+105.i)T+(−5.54e3+4.03e3i)T2 1 + (34.2 + 105. i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+12.1T+1.21e4T2 1 + 12.1T + 1.21e4T^{2}
29 1+(86.7−266.i)T+(−1.97e4−1.43e4i)T2 1 + (86.7 - 266. i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(74.0−53.8i)T+(9.20e3−2.83e4i)T2 1 + (74.0 - 53.8i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(86.7−266.i)T+(−4.09e4−2.97e4i)T2 1 + (86.7 - 266. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(1.85+5.69i)T+(−5.57e4+4.05e4i)T2 1 + (1.85 + 5.69i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+126.T+7.95e4T2 1 + 126.T + 7.95e4T^{2}
47 1+(72.3+222.i)T+(−8.39e4+6.10e4i)T2 1 + (72.3 + 222. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−7.78+5.65i)T+(4.60e4−1.41e5i)T2 1 + (-7.78 + 5.65i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−96.3+296.i)T+(−1.66e5−1.20e5i)T2 1 + (-96.3 + 296. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−719.−522.i)T+(7.01e4+2.15e5i)T2 1 + (-719. - 522. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1+589.T+3.00e5T2 1 + 589.T + 3.00e5T^{2}
71 1+(−212.−154.i)T+(1.10e5+3.40e5i)T2 1 + (-212. - 154. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(52.0−160.i)T+(−3.14e5−2.28e5i)T2 1 + (52.0 - 160. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(403.−293.i)T+(1.52e5−4.68e5i)T2 1 + (403. - 293. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−261.−189.i)T+(1.76e5+5.43e5i)T2 1 + (-261. - 189. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−1.18e3T+7.04e5T2 1 - 1.18e3T + 7.04e5T^{2}
97 1+(−938.+681.i)T+(2.82e5−8.68e5i)T2 1 + (-938. + 681. 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

−14.46569602981937280913831630172, −13.01340141339736584814174089465, −11.73015033078859829204393207494, −10.34039652923989183932413373536, −9.338825793739287907168421075607, −8.786488393612752713343396208570, −7.11053862128640100635819867727, −6.63671999465136622552868569752, −4.98656730002981875769803147028, −2.24097078285833533721395480699, 0.28584594916189132593218092620, 1.97714137404701344110924073940, 3.76767045326732602545158903797, 5.91755753561279760710302660067, 7.62096144966296851485117458866, 8.626142435531437543544707598449, 9.688206033008023591943732745942, 10.50509442900965175839635078465, 11.35608081568487536903833395814, 12.62254198293903332881933434663

Graph of the ZZ-function along the critical line