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Properties

Label 2-720-5.4-c3-0-17
Degree 22
Conductor 720720
Sign 0.178−0.983i0.178 - 0.983i
Analytic cond. 42.481342.4813
Root an. cond. 6.517776.51777
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  + (−2 + 11i)5-s − 2i·7-s + 70·11-s + 54i·13-s − 22i·17-s + 24·19-s − 100i·23-s + (−117 − 44i)25-s + 216·29-s − 208·31-s + (22 + 4i)35-s + 254i·37-s + 206·41-s − 292i·43-s + 320i·47-s + ⋯
L(s)  = 1  + (−0.178 + 0.983i)5-s − 0.107i·7-s + 1.91·11-s + 1.15i·13-s − 0.313i·17-s + 0.289·19-s − 0.906i·23-s + (−0.936 − 0.351i)25-s + 1.38·29-s − 1.20·31-s + (0.106 + 0.0193i)35-s + 1.12i·37-s + 0.784·41-s − 1.03i·43-s + 0.993i·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.178−0.983i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+3/2) L(s)=((0.178−0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.178−0.983i0.178 - 0.983i
Analytic conductor: 42.481342.4813
Root analytic conductor: 6.517776.51777
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ720(289,⋅)\chi_{720} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :3/2), 0.178−0.983i)(2,\ 720,\ (\ :3/2),\ 0.178 - 0.983i)

Particular Values

L(2)L(2) ≈\approx 2.0751683282.075168328
L(12)L(\frac12) ≈\approx 2.0751683282.075168328
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
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

−10.21494300752468365668362930674, −9.316450769193020078576240448388, −8.653004967483442220759748084364, −7.31706722832136606978202817879, −6.73957078026412521026314925313, −6.06182631274567230773366555169, −4.46593762602424361006896742301, −3.77381393481421268474641196108, −2.54958695918014148132565777785, −1.20951081433644063263851661442, 0.66583347662454195314243970244, 1.66833690159052246119388559745, 3.39744716806499982102542324336, 4.23169931365293033787417860015, 5.34157028449350730733491070418, 6.15067808331699079056410598138, 7.28970608964527741928219389677, 8.200948299399744884715549749707, 9.048788365532477163244870384900, 9.568100281711732189548584610546

Graph of the ZZ-function along the critical line