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Properties

Label 2-720-20.19-c2-0-28
Degree 22
Conductor 720720
Sign −0.399+0.916i-0.399 + 0.916i
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 − 4.58i)5-s + 3.46·7-s − 15.8i·11-s − 9.16i·13-s − 9.16i·17-s + 31.7i·19-s − 27.7·23-s + (−17 − 18.3i)25-s + 8·29-s + (6.92 − 15.8i)35-s − 45.8i·37-s − 50·41-s + 62.3·43-s + 48.4·47-s − 37·49-s + ⋯
L(s)  = 1  + (0.400 − 0.916i)5-s + 0.494·7-s − 1.44i·11-s − 0.705i·13-s − 0.539i·17-s + 1.67i·19-s − 1.20·23-s + (−0.680 − 0.733i)25-s + 0.275·29-s + (0.197 − 0.453i)35-s − 1.23i·37-s − 1.21·41-s + 1.45·43-s + 1.03·47-s − 0.755·49-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.399+0.916i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((−0.399+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.399+0.916i-0.399 + 0.916i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(559,⋅)\chi_{720} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), −0.399+0.916i)(2,\ 720,\ (\ :1),\ -0.399 + 0.916i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.6717060771.671706077
L(12)L(\frac12) ≈\approx 1.6717060771.671706077
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+4.58i)T 1 + (-2 + 4.58i)T
good7 1−3.46T+49T2 1 - 3.46T + 49T^{2}
11 1+15.8iT−121T2 1 + 15.8iT - 121T^{2}
13 1+9.16iT−169T2 1 + 9.16iT - 169T^{2}
17 1+9.16iT−289T2 1 + 9.16iT - 289T^{2}
19 1−31.7iT−361T2 1 - 31.7iT - 361T^{2}
23 1+27.7T+529T2 1 + 27.7T + 529T^{2}
29 1−8T+841T2 1 - 8T + 841T^{2}
31 1−961T2 1 - 961T^{2}
37 1+45.8iT−1.36e3T2 1 + 45.8iT - 1.36e3T^{2}
41 1+50T+1.68e3T2 1 + 50T + 1.68e3T^{2}
43 1−62.3T+1.84e3T2 1 - 62.3T + 1.84e3T^{2}
47 1−48.4T+2.20e3T2 1 - 48.4T + 2.20e3T^{2}
53 1+27.4iT−2.80e3T2 1 + 27.4iT - 2.80e3T^{2}
59 1−15.8iT−3.48e3T2 1 - 15.8iT - 3.48e3T^{2}
61 1+26T+3.72e3T2 1 + 26T + 3.72e3T^{2}
67 1+55.4T+4.48e3T2 1 + 55.4T + 4.48e3T^{2}
71 1−95.2iT−5.04e3T2 1 - 95.2iT - 5.04e3T^{2}
73 1+128.iT−5.32e3T2 1 + 128. iT - 5.32e3T^{2}
79 1+126.iT−6.24e3T2 1 + 126. iT - 6.24e3T^{2}
83 1+131.T+6.88e3T2 1 + 131.T + 6.88e3T^{2}
89 1−86T+7.92e3T2 1 - 86T + 7.92e3T^{2}
97 1+109.iT−9.40e3T2 1 + 109. iT - 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

−9.984983273489167289034549536783, −8.933195466043008725147376520206, −8.281605107383823189200707826579, −7.60443320646196760613427460405, −5.91701893899739685741103807724, −5.70167509585815542325247477659, −4.45390018446144366138502426902, −3.35559592860417335864960092190, −1.85136119243904226762972696511, −0.57535073069386502771676904811, 1.74411503909008007444689376533, 2.63900405355367211950264698925, 4.11122302901558982019052664272, 4.95540754247163696830167759984, 6.23713520733827214269704918690, 6.96463973678336517234389521577, 7.70302430309216305955090543030, 8.872181714388723811909978852215, 9.754738066679740224949796441472, 10.39790646529618695443765712504

Graph of the ZZ-function along the critical line