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Properties

Label 2-880-5.4-c3-0-86
Degree 22
Conductor 880880
Sign −0.223−0.974i-0.223 - 0.974i
Analytic cond. 51.921651.9216
Root an. cond. 7.205677.20567
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.35i·3-s + (−2.5 − 10.8i)5-s − 8.71i·7-s + 7.99·9-s + 11·11-s + 69.7i·13-s + (−47.5 + 10.8i)15-s + 26.1i·17-s − 68·19-s − 38.0·21-s − 117. i·23-s + (−112. + 54.4i)25-s − 152. i·27-s − 260·29-s − 175·31-s + ⋯
L(s)  = 1  − 0.838i·3-s + (−0.223 − 0.974i)5-s − 0.470i·7-s + 0.296·9-s + 0.301·11-s + 1.48i·13-s + (−0.817 + 0.187i)15-s + 0.373i·17-s − 0.821·19-s − 0.394·21-s − 1.06i·23-s + (−0.900 + 0.435i)25-s − 1.08i·27-s − 1.66·29-s − 1.01·31-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=((−0.223−0.974i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+3/2) L(s)=((−0.223−0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −0.223−0.974i-0.223 - 0.974i
Analytic conductor: 51.921651.9216
Root analytic conductor: 7.205677.20567
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ880(529,⋅)\chi_{880} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 880, ( :3/2), −0.223−0.974i)(2,\ 880,\ (\ :3/2),\ -0.223 - 0.974i)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1
5 1+(2.5+10.8i)T 1 + (2.5 + 10.8i)T
11 1−11T 1 - 11T
good3 1+4.35iT−27T2 1 + 4.35iT - 27T^{2}
7 1+8.71iT−343T2 1 + 8.71iT - 343T^{2}
13 1−69.7iT−2.19e3T2 1 - 69.7iT - 2.19e3T^{2}
17 1−26.1iT−4.91e3T2 1 - 26.1iT - 4.91e3T^{2}
19 1+68T+6.85e3T2 1 + 68T + 6.85e3T^{2}
23 1+117.iT−1.21e4T2 1 + 117. iT - 1.21e4T^{2}
29 1+260T+2.43e4T2 1 + 260T + 2.43e4T^{2}
31 1+175T+2.97e4T2 1 + 175T + 2.97e4T^{2}
37 1−169.iT−5.06e4T2 1 - 169. iT - 5.06e4T^{2}
41 1+380T+6.89e4T2 1 + 380T + 6.89e4T^{2}
43 1+305.iT−7.95e4T2 1 + 305. iT - 7.95e4T^{2}
47 1−305.iT−1.03e5T2 1 - 305. iT - 1.03e5T^{2}
53 1−453.iT−1.48e5T2 1 - 453. iT - 1.48e5T^{2}
59 1+143T+2.05e5T2 1 + 143T + 2.05e5T^{2}
61 1−676T+2.26e5T2 1 - 676T + 2.26e5T^{2}
67 1−527.iT−3.00e5T2 1 - 527. iT - 3.00e5T^{2}
71 1+1.03e3T+3.57e5T2 1 + 1.03e3T + 3.57e5T^{2}
73 1−331.iT−3.89e5T2 1 - 331. iT - 3.89e5T^{2}
79 1−218T+4.93e5T2 1 - 218T + 4.93e5T^{2}
83 1−758.iT−5.71e5T2 1 - 758. iT - 5.71e5T^{2}
89 1+1.27e3T+7.04e5T2 1 + 1.27e3T + 7.04e5T^{2}
97 1+771.iT−9.12e5T2 1 + 771. iT - 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

−8.973524909901191267283101494660, −8.368820472229602627060790169402, −7.29532797248977256757604664800, −6.78587756900720496850875367245, −5.74094225630581476089605988780, −4.42910729672148912344157577890, −3.95996554141189102432383437016, −2.03778139979098331335060036385, −1.32368266717115646098174359939, 0, 1.96143407284494698426627859229, 3.28368367897313935058429439411, 3.82825849100502005167608208822, 5.14681888173533660295886295544, 5.87160098364068095329608726569, 7.05289686269306588950857296615, 7.72136227794565121108694769649, 8.820366721960896394006387520889, 9.685802593369645885903864558656

Graph of the ZZ-function along the critical line