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Properties

Label 2-92-92.19-c1-0-0
Degree 22
Conductor 9292
Sign −0.395−0.918i-0.395 - 0.918i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
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  + (−1.29 + 0.579i)2-s + (0.597 + 2.03i)3-s + (1.32 − 1.49i)4-s + (−1.54 + 2.40i)5-s + (−1.94 − 2.27i)6-s + (−0.249 − 1.73i)7-s + (−0.847 + 2.69i)8-s + (−1.25 + 0.806i)9-s + (0.599 − 3.99i)10-s + (−0.620 + 1.35i)11-s + (3.83 + 1.80i)12-s + (−0.727 + 5.05i)13-s + (1.32 + 2.09i)14-s + (−5.80 − 1.70i)15-s + (−0.470 − 3.97i)16-s + (4.17 − 3.61i)17-s + ⋯
L(s)  = 1  + (−0.912 + 0.409i)2-s + (0.344 + 1.17i)3-s + (0.664 − 0.747i)4-s + (−0.690 + 1.07i)5-s + (−0.795 − 0.929i)6-s + (−0.0943 − 0.656i)7-s + (−0.299 + 0.954i)8-s + (−0.418 + 0.268i)9-s + (0.189 − 1.26i)10-s + (−0.187 + 0.409i)11-s + (1.10 + 0.522i)12-s + (−0.201 + 1.40i)13-s + (0.355 + 0.560i)14-s + (−1.49 − 0.440i)15-s + (−0.117 − 0.993i)16-s + (1.01 − 0.877i)17-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((−0.395−0.918i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((−0.395−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: −0.395−0.918i-0.395 - 0.918i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ92(19,⋅)\chi_{92} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :1/2), −0.395−0.918i)(2,\ 92,\ (\ :1/2),\ -0.395 - 0.918i)

Particular Values

L(1)L(1) ≈\approx 0.371671+0.564842i0.371671 + 0.564842i
L(12)L(\frac12) ≈\approx 0.371671+0.564842i0.371671 + 0.564842i
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)
bad2 1+(1.29−0.579i)T 1 + (1.29 - 0.579i)T
23 1+(−4.76+0.575i)T 1 + (-4.76 + 0.575i)T
good3 1+(−0.597−2.03i)T+(−2.52+1.62i)T2 1 + (-0.597 - 2.03i)T + (-2.52 + 1.62i)T^{2}
5 1+(1.54−2.40i)T+(−2.07−4.54i)T2 1 + (1.54 - 2.40i)T + (-2.07 - 4.54i)T^{2}
7 1+(0.249+1.73i)T+(−6.71+1.97i)T2 1 + (0.249 + 1.73i)T + (-6.71 + 1.97i)T^{2}
11 1+(0.620−1.35i)T+(−7.20−8.31i)T2 1 + (0.620 - 1.35i)T + (-7.20 - 8.31i)T^{2}
13 1+(0.727−5.05i)T+(−12.4−3.66i)T2 1 + (0.727 - 5.05i)T + (-12.4 - 3.66i)T^{2}
17 1+(−4.17+3.61i)T+(2.41−16.8i)T2 1 + (-4.17 + 3.61i)T + (2.41 - 16.8i)T^{2}
19 1+(−4.01+4.63i)T+(−2.70−18.8i)T2 1 + (-4.01 + 4.63i)T + (-2.70 - 18.8i)T^{2}
29 1+(4.51+5.21i)T+(−4.12+28.7i)T2 1 + (4.51 + 5.21i)T + (-4.12 + 28.7i)T^{2}
31 1+(0.376−1.28i)T+(−26.0−16.7i)T2 1 + (0.376 - 1.28i)T + (-26.0 - 16.7i)T^{2}
37 1+(−1.44−2.25i)T+(−15.3+33.6i)T2 1 + (-1.44 - 2.25i)T + (-15.3 + 33.6i)T^{2}
41 1+(0.151+0.0973i)T+(17.0+37.2i)T2 1 + (0.151 + 0.0973i)T + (17.0 + 37.2i)T^{2}
43 1+(3.19−0.936i)T+(36.1−23.2i)T2 1 + (3.19 - 0.936i)T + (36.1 - 23.2i)T^{2}
47 1+0.713iT−47T2 1 + 0.713iT - 47T^{2}
53 1+(7.04−1.01i)T+(50.8−14.9i)T2 1 + (7.04 - 1.01i)T + (50.8 - 14.9i)T^{2}
59 1+(9.95+1.43i)T+(56.6+16.6i)T2 1 + (9.95 + 1.43i)T + (56.6 + 16.6i)T^{2}
61 1+(−3.77+12.8i)T+(−51.3−32.9i)T2 1 + (-3.77 + 12.8i)T + (-51.3 - 32.9i)T^{2}
67 1+(1.87+4.10i)T+(−43.8+50.6i)T2 1 + (1.87 + 4.10i)T + (-43.8 + 50.6i)T^{2}
71 1+(9.80−4.47i)T+(46.4−53.6i)T2 1 + (9.80 - 4.47i)T + (46.4 - 53.6i)T^{2}
73 1+(3.48−4.02i)T+(−10.3−72.2i)T2 1 + (3.48 - 4.02i)T + (-10.3 - 72.2i)T^{2}
79 1+(0.274−1.90i)T+(−75.7−22.2i)T2 1 + (0.274 - 1.90i)T + (-75.7 - 22.2i)T^{2}
83 1+(−11.1+7.16i)T+(34.4−75.4i)T2 1 + (-11.1 + 7.16i)T + (34.4 - 75.4i)T^{2}
89 1+(0.565+1.92i)T+(−74.8+48.1i)T2 1 + (0.565 + 1.92i)T + (-74.8 + 48.1i)T^{2}
97 1+(4.90−7.63i)T+(−40.2−88.2i)T2 1 + (4.90 - 7.63i)T + (-40.2 - 88.2i)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

−14.75509129131818319069842560395, −13.94134275259853294701835663111, −11.64636660078885916281773697447, −10.92953642565772226027915909224, −9.850620287934558267306234356124, −9.236932284100426323623265629730, −7.53235755593215071199129604199, −6.88322120560703692243302112349, −4.76733977193908491006338369824, −3.15886656884033572209630736201, 1.22580483711375761620590504139, 3.20796711091308429044459808791, 5.68557289030257547513613316641, 7.56056689712165755404905975735, 8.058585753060439281451290669475, 8.999774624009873796776269689379, 10.44824352195829587557608324598, 11.94836156969920890003918364394, 12.53563950388978168895523634928, 13.12706508367529127008121375716

Graph of the ZZ-function along the critical line