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Properties

Label 2-720-4.3-c2-0-15
Degree 22
Conductor 720720
Sign −0.5+0.866i-0.5 + 0.866i
Analytic cond. 19.618519.6185
Root an. cond. 4.429284.42928
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·5-s + 9.06i·7-s − 4.28i·11-s − 9.41·13-s − 18·17-s − 36.2i·19-s + 22.9i·23-s + 5.00·25-s + 44.8·29-s − 35.2i·31-s − 20.2i·35-s + 6.58·37-s − 52.2·41-s − 28.8i·43-s − 90.1i·47-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29i·7-s − 0.389i·11-s − 0.724·13-s − 1.05·17-s − 1.90i·19-s + 0.996i·23-s + 0.200·25-s + 1.54·29-s − 1.13i·31-s − 0.579i·35-s + 0.177·37-s − 1.27·41-s − 0.670i·43-s − 1.91i·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.5+0.866i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((−0.5+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.5+0.866i-0.5 + 0.866i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(271,⋅)\chi_{720} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), −0.5+0.866i)(2,\ 720,\ (\ :1),\ -0.5 + 0.866i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.54750016240.5475001624
L(12)L(\frac12) ≈\approx 0.54750016240.5475001624
L(2)L(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+2.23T 1 + 2.23T
good7 1−9.06iT−49T2 1 - 9.06iT - 49T^{2}
11 1+4.28iT−121T2 1 + 4.28iT - 121T^{2}
13 1+9.41T+169T2 1 + 9.41T + 169T^{2}
17 1+18T+289T2 1 + 18T + 289T^{2}
19 1+36.2iT−361T2 1 + 36.2iT - 361T^{2}
23 1−22.9iT−529T2 1 - 22.9iT - 529T^{2}
29 1−44.8T+841T2 1 - 44.8T + 841T^{2}
31 1+35.2iT−961T2 1 + 35.2iT - 961T^{2}
37 1−6.58T+1.36e3T2 1 - 6.58T + 1.36e3T^{2}
41 1+52.2T+1.68e3T2 1 + 52.2T + 1.68e3T^{2}
43 1+28.8iT−1.84e3T2 1 + 28.8iT - 1.84e3T^{2}
47 1+90.1iT−2.20e3T2 1 + 90.1iT - 2.20e3T^{2}
53 1+52.2T+2.80e3T2 1 + 52.2T + 2.80e3T^{2}
59 1+17.1iT−3.48e3T2 1 + 17.1iT - 3.48e3T^{2}
61 1+50.5T+3.72e3T2 1 + 50.5T + 3.72e3T^{2}
67 1+33.1iT−4.48e3T2 1 + 33.1iT - 4.48e3T^{2}
71 1+20.1iT−5.04e3T2 1 + 20.1iT - 5.04e3T^{2}
73 1+91.6T+5.32e3T2 1 + 91.6T + 5.32e3T^{2}
79 1+42.8iT−6.24e3T2 1 + 42.8iT - 6.24e3T^{2}
83 1−22.3iT−6.88e3T2 1 - 22.3iT - 6.88e3T^{2}
89 1+47.6T+7.92e3T2 1 + 47.6T + 7.92e3T^{2}
97 1+160.T+9.40e3T2 1 + 160.T + 9.40e3T^{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

−9.777001380695223774437175726667, −8.946271507237837294037211348633, −8.416027863237256524158062977361, −7.24826059709875262024955942401, −6.42833519498707328197249739787, −5.30689146390029485069246659303, −4.55015624217656607730695593699, −3.08818026969010172978020151958, −2.19923826177031711785418606456, −0.19381442377873155802409579160, 1.38179626340448308843889577599, 2.97123484421082209794274889103, 4.20874931189570728910642451186, 4.73259953585066063772744938246, 6.28100680247349600879873704327, 7.04360105212120725429100136339, 7.85879684291984121109542862582, 8.627020564082968737503518958073, 9.931575698883027979388856226885, 10.37103574611051485180370108688

Graph of the ZZ-function along the critical line