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Properties

Label 2-800-5.4-c1-0-10
Degree 22
Conductor 800800
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 6.388036.38803
Root an. cond. 2.527452.52745
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  − 2.23i·3-s + 4.47i·7-s − 2.00·9-s + 2.23·11-s − 4i·13-s − 7i·17-s + 6.70·19-s + 10.0·21-s + 4.47i·23-s − 2.23i·27-s + 4.47·31-s − 5.00i·33-s + 2i·37-s − 8.94·39-s + 5·41-s + ⋯
L(s)  = 1  − 1.29i·3-s + 1.69i·7-s − 0.666·9-s + 0.674·11-s − 1.10i·13-s − 1.69i·17-s + 1.53·19-s + 2.18·21-s + 0.932i·23-s − 0.430i·27-s + 0.803·31-s − 0.870i·33-s + 0.328i·37-s − 1.43·39-s + 0.780·41-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 6.388036.38803
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ800(449,⋅)\chi_{800} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :1/2), 0.447+0.894i)(2,\ 800,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.39974−0.865087i1.39974 - 0.865087i
L(12)L(\frac12) ≈\approx 1.39974−0.865087i1.39974 - 0.865087i
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
5 1 1
good3 1+2.23iT−3T2 1 + 2.23iT - 3T^{2}
7 1−4.47iT−7T2 1 - 4.47iT - 7T^{2}
11 1−2.23T+11T2 1 - 2.23T + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1+7iT−17T2 1 + 7iT - 17T^{2}
19 1−6.70T+19T2 1 - 6.70T + 19T^{2}
23 1−4.47iT−23T2 1 - 4.47iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−4.47T+31T2 1 - 4.47T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−5T+41T2 1 - 5T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+8.94iT−47T2 1 + 8.94iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+8.94T+59T2 1 + 8.94T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+2.23iT−67T2 1 + 2.23iT - 67T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1−9iT−73T2 1 - 9iT - 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−11.1iT−83T2 1 - 11.1iT - 83T^{2}
89 1−5T+89T2 1 - 5T + 89T^{2}
97 1−2iT−97T2 1 - 2iT - 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

−9.839803862765569531545634382689, −9.240965149016500304891626780701, −8.272208392090129247198265658576, −7.53113146570887509660281653628, −6.71644854216888003802266110682, −5.70754864344051235542333078989, −5.13483204895572045308980607817, −3.19190200246757812188398439821, −2.37203888630317906529697764637, −1.01332156552798810380254807772, 1.32094182663328476614322895001, 3.35700141286033908289106089576, 4.21308895363582647916284479630, 4.53474837684645232812174177162, 6.03123191644873442223989771963, 6.95284185642370767670513200014, 7.85996354933156287370130199078, 9.009461987044900305652601609316, 9.682115219200449726619064213667, 10.43617957435955947721893977917

Graph of the ZZ-function along the critical line