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Properties

Label 2-90-5.4-c3-0-1
Degree 22
Conductor 9090
Sign −0.983−0.178i-0.983 - 0.178i
Analytic cond. 5.310175.31017
Root an. cond. 2.304382.30438
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  + 2i·2-s − 4·4-s + (−2 + 11i)5-s + 2i·7-s − 8i·8-s + (−22 − 4i)10-s − 70·11-s + 54i·13-s − 4·14-s + 16·16-s − 22i·17-s − 24·19-s + (8 − 44i)20-s − 140i·22-s + 100i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.178 + 0.983i)5-s + 0.107i·7-s − 0.353i·8-s + (−0.695 − 0.126i)10-s − 1.91·11-s + 1.15i·13-s − 0.0763·14-s + 0.250·16-s − 0.313i·17-s − 0.289·19-s + (0.0894 − 0.491i)20-s − 1.35i·22-s + 0.906i·23-s + ⋯

Functional equation

Λ(s)=(90s/2 ΓC(s) L(s)=((−0.983−0.178i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(90s/2 ΓC(s+3/2) L(s)=((−0.983−0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: −0.983−0.178i-0.983 - 0.178i
Analytic conductor: 5.310175.31017
Root analytic conductor: 2.304382.30438
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ90(19,⋅)\chi_{90} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :3/2), −0.983−0.178i)(2,\ 90,\ (\ :3/2),\ -0.983 - 0.178i)

Particular Values

L(2)L(2) ≈\approx 0.0798739+0.885815i0.0798739 + 0.885815i
L(12)L(\frac12) ≈\approx 0.0798739+0.885815i0.0798739 + 0.885815i
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)
bad2 1−2iT 1 - 2iT
3 1 1
5 1+(2−11i)T 1 + (2 - 11i)T
good7 1−2iT−343T2 1 - 2iT - 343T^{2}
11 1+70T+1.33e3T2 1 + 70T + 1.33e3T^{2}
13 1−54iT−2.19e3T2 1 - 54iT - 2.19e3T^{2}
17 1+22iT−4.91e3T2 1 + 22iT - 4.91e3T^{2}
19 1+24T+6.85e3T2 1 + 24T + 6.85e3T^{2}
23 1−100iT−1.21e4T2 1 - 100iT - 1.21e4T^{2}
29 1−216T+2.43e4T2 1 - 216T + 2.43e4T^{2}
31 1−208T+2.97e4T2 1 - 208T + 2.97e4T^{2}
37 1−254iT−5.06e4T2 1 - 254iT - 5.06e4T^{2}
41 1−206T+6.89e4T2 1 - 206T + 6.89e4T^{2}
43 1−292iT−7.95e4T2 1 - 292iT - 7.95e4T^{2}
47 1+320iT−1.03e5T2 1 + 320iT - 1.03e5T^{2}
53 1−402iT−1.48e5T2 1 - 402iT - 1.48e5T^{2}
59 1+370T+2.05e5T2 1 + 370T + 2.05e5T^{2}
61 1+550T+2.26e5T2 1 + 550T + 2.26e5T^{2}
67 1+728iT−3.00e5T2 1 + 728iT - 3.00e5T^{2}
71 1−540T+3.57e5T2 1 - 540T + 3.57e5T^{2}
73 1−604iT−3.89e5T2 1 - 604iT - 3.89e5T^{2}
79 1+792T+4.93e5T2 1 + 792T + 4.93e5T^{2}
83 1+404iT−5.71e5T2 1 + 404iT - 5.71e5T^{2}
89 1+938T+7.04e5T2 1 + 938T + 7.04e5T^{2}
97 1+56iT−9.12e5T2 1 + 56iT - 9.12e5T^{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

−14.10781147973330493133395981977, −13.44013595304228410489311100526, −11.99168109912117979018073388800, −10.76332721886548661014504211752, −9.778562641608067821768298416545, −8.250181407692054391244660353010, −7.28779139547374256604122749654, −6.14531695116063254432991272069, −4.67076659534695145022232315746, −2.79227728644389770554895895844, 0.51702683540305336395139394427, 2.68573485585703268549144005533, 4.52716288594949031469371517418, 5.61540191644989400971545842340, 7.83827582872334273059172737378, 8.578051630475317594238774977570, 10.10642314008639750098482114518, 10.78724315461149715897162670577, 12.35502594416933827554039524549, 12.83794671659031800713798951246

Graph of the ZZ-function along the critical line