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Properties

Label 2-6000-5.4-c1-0-70
Degree 22
Conductor 60006000
Sign ii
Analytic cond. 47.910247.9102
Root an. cond. 6.921726.92172
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  − i·3-s + 4.73i·7-s − 9-s + 4.34·11-s − 2.30i·13-s + 0.618i·17-s − 4.41·19-s + 4.73·21-s − 6.34i·23-s + i·27-s − 7.54·29-s − 4.80·31-s − 4.34i·33-s − 10.9i·37-s − 2.30·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.78i·7-s − 0.333·9-s + 1.31·11-s − 0.639i·13-s + 0.149i·17-s − 1.01·19-s + 1.03·21-s − 1.32i·23-s + 0.192i·27-s − 1.40·29-s − 0.862·31-s − 0.757i·33-s − 1.80i·37-s − 0.369·39-s + ⋯

Functional equation

Λ(s)=(6000s/2 ΓC(s) L(s)=(i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6000s/2 ΓC(s+1/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60006000    =    24⋅3⋅532^{4} \cdot 3 \cdot 5^{3}
Sign: ii
Analytic conductor: 47.910247.9102
Root analytic conductor: 6.921726.92172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6000(1249,⋅)\chi_{6000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6000, ( :1/2), i)(2,\ 6000,\ (\ :1/2),\ i)

Particular Values

L(1)L(1) ≈\approx 1.3741916011.374191601
L(12)L(\frac12) ≈\approx 1.3741916011.374191601
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
3 1+iT 1 + iT
5 1 1
good7 1−4.73iT−7T2 1 - 4.73iT - 7T^{2}
11 1−4.34T+11T2 1 - 4.34T + 11T^{2}
13 1+2.30iT−13T2 1 + 2.30iT - 13T^{2}
17 1−0.618iT−17T2 1 - 0.618iT - 17T^{2}
19 1+4.41T+19T2 1 + 4.41T + 19T^{2}
23 1+6.34iT−23T2 1 + 6.34iT - 23T^{2}
29 1+7.54T+29T2 1 + 7.54T + 29T^{2}
31 1+4.80T+31T2 1 + 4.80T + 31T^{2}
37 1+10.9iT−37T2 1 + 10.9iT - 37T^{2}
41 1+5.20T+41T2 1 + 5.20T + 41T^{2}
43 1+6.85iT−43T2 1 + 6.85iT - 43T^{2}
47 1+0.165iT−47T2 1 + 0.165iT - 47T^{2}
53 1−2.16iT−53T2 1 - 2.16iT - 53T^{2}
59 1−12.9T+59T2 1 - 12.9T + 59T^{2}
61 1−4.11T+61T2 1 - 4.11T + 61T^{2}
67 1+14.6iT−67T2 1 + 14.6iT - 67T^{2}
71 1−4.92T+71T2 1 - 4.92T + 71T^{2}
73 1−7.91iT−73T2 1 - 7.91iT - 73T^{2}
79 1−17.0T+79T2 1 - 17.0T + 79T^{2}
83 1−5.72iT−83T2 1 - 5.72iT - 83T^{2}
89 1+0.816T+89T2 1 + 0.816T + 89T^{2}
97 1+7.20iT−97T2 1 + 7.20iT - 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

−8.089358623624594872353466465524, −7.02976732694934975626017225436, −6.50540050809015695334060862176, −5.71628195765955340129685397085, −5.36999605552138083515376080420, −4.16139647712172100846634689232, −3.37653686068264132714555389966, −2.23202021655212948037062778437, −1.92954107625152772597088603176, −0.37256297523656360201148427452, 1.05984166767295396924635449964, 1.90832263229045716149693332363, 3.49570423609070497626109344516, 3.79477051494600534808891857336, 4.44331320456036136736382275229, 5.23171415651609678919225846582, 6.33641124724757154733829165490, 6.82242372797424890503421928186, 7.45726976621930356317286583659, 8.249287434840042760925624310520

Graph of the ZZ-function along the critical line