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Properties

Label 2-189-21.20-c1-0-1
Degree 22
Conductor 189189
Sign −0.377−0.925i-0.377 - 0.925i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
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  + 1.41i·2-s − 1.73·5-s + (1 + 2.44i)7-s + 2.82i·8-s − 2.44i·10-s + 1.41i·11-s + 2.44i·13-s + (−3.46 + 1.41i)14-s − 4.00·16-s + 5.19·17-s − 7.34i·19-s − 2.00·22-s − 2.82i·23-s − 2.00·25-s − 3.46·26-s + ⋯
L(s)  = 1  + 0.999i·2-s − 0.774·5-s + (0.377 + 0.925i)7-s + 0.999i·8-s − 0.774i·10-s + 0.426i·11-s + 0.679i·13-s + (−0.925 + 0.377i)14-s − 1.00·16-s + 1.26·17-s − 1.68i·19-s − 0.426·22-s − 0.589i·23-s − 0.400·25-s − 0.679·26-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.377−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((−0.377−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.377−0.925i-0.377 - 0.925i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(188,⋅)\chi_{189} (188, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), −0.377−0.925i)(2,\ 189,\ (\ :1/2),\ -0.377 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.658358+0.979882i0.658358 + 0.979882i
L(12)L(\frac12) ≈\approx 0.658358+0.979882i0.658358 + 0.979882i
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
7 1+(−1−2.44i)T 1 + (-1 - 2.44i)T
good2 1−1.41iT−2T2 1 - 1.41iT - 2T^{2}
5 1+1.73T+5T2 1 + 1.73T + 5T^{2}
11 1−1.41iT−11T2 1 - 1.41iT - 11T^{2}
13 1−2.44iT−13T2 1 - 2.44iT - 13T^{2}
17 1−5.19T+17T2 1 - 5.19T + 17T^{2}
19 1+7.34iT−19T2 1 + 7.34iT - 19T^{2}
23 1+2.82iT−23T2 1 + 2.82iT - 23T^{2}
29 1+7.07iT−29T2 1 + 7.07iT - 29T^{2}
31 1−2.44iT−31T2 1 - 2.44iT - 31T^{2}
37 1−5T+37T2 1 - 5T + 37T^{2}
41 1−8.66T+41T2 1 - 8.66T + 41T^{2}
43 1−5T+43T2 1 - 5T + 43T^{2}
47 1+8.66T+47T2 1 + 8.66T + 47T^{2}
53 1+11.3iT−53T2 1 + 11.3iT - 53T^{2}
59 1−8.66T+59T2 1 - 8.66T + 59T^{2}
61 1+2.44iT−61T2 1 + 2.44iT - 61T^{2}
67 1−2T+67T2 1 - 2T + 67T^{2}
71 1−14.1iT−71T2 1 - 14.1iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+13T+79T2 1 + 13T + 79T^{2}
83 1−1.73T+83T2 1 - 1.73T + 83T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+17.1iT−97T2 1 + 17.1iT - 97T^{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

−12.83217219130144172886084074141, −11.67120188013919176218549325931, −11.33486793844748845137857339627, −9.639994425197906743321190836037, −8.507092058131594793711533661069, −7.72969249655452372306215068052, −6.75682140315435819717536517767, −5.60233471199409282489903453001, −4.47567587387250422591413427531, −2.50285706865570168890423261532, 1.21390625503354902557172281617, 3.26072537580287945895711806105, 4.06322499893154788370364151799, 5.81826367315714878428283556666, 7.42781759225687197034043682863, 8.006987996081415846114091344901, 9.700248769112340950507436858839, 10.52010450562826378801543801606, 11.25401025099405174551874117609, 12.13141041694563760704111852812

Graph of the ZZ-function along the critical line