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Properties

Label 2-720-20.19-c2-0-3
Degree 22
Conductor 720720
Sign −0.919−0.392i-0.919 - 0.392i
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  + (−4 + 3i)5-s + 3.46·7-s − 3.46i·11-s − 6i·13-s + 18i·17-s + 13.8i·19-s − 6.92·23-s + (7 − 24i)25-s − 28·29-s + 48.4i·31-s + (−13.8 + 10.3i)35-s − 30i·37-s − 2·41-s − 62.3·43-s − 55.4·47-s + ⋯
L(s)  = 1  + (−0.800 + 0.600i)5-s + 0.494·7-s − 0.314i·11-s − 0.461i·13-s + 1.05i·17-s + 0.729i·19-s − 0.301·23-s + (0.280 − 0.959i)25-s − 0.965·29-s + 1.56i·31-s + (−0.395 + 0.296i)35-s − 0.810i·37-s − 0.0487·41-s − 1.45·43-s − 1.17·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.919−0.392i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((−0.919−0.392i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.919−0.392i-0.919 - 0.392i
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), −0.919−0.392i)(2,\ 720,\ (\ :1),\ -0.919 - 0.392i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.56469940180.5646994018
L(12)L(\frac12) ≈\approx 0.56469940180.5646994018
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+(4−3i)T 1 + (4 - 3i)T
good7 1−3.46T+49T2 1 - 3.46T + 49T^{2}
11 1+3.46iT−121T2 1 + 3.46iT - 121T^{2}
13 1+6iT−169T2 1 + 6iT - 169T^{2}
17 1−18iT−289T2 1 - 18iT - 289T^{2}
19 1−13.8iT−361T2 1 - 13.8iT - 361T^{2}
23 1+6.92T+529T2 1 + 6.92T + 529T^{2}
29 1+28T+841T2 1 + 28T + 841T^{2}
31 1−48.4iT−961T2 1 - 48.4iT - 961T^{2}
37 1+30iT−1.36e3T2 1 + 30iT - 1.36e3T^{2}
41 1+2T+1.68e3T2 1 + 2T + 1.68e3T^{2}
43 1+62.3T+1.84e3T2 1 + 62.3T + 1.84e3T^{2}
47 1+55.4T+2.20e3T2 1 + 55.4T + 2.20e3T^{2}
53 1+102iT−2.80e3T2 1 + 102iT - 2.80e3T^{2}
59 1−100.iT−3.48e3T2 1 - 100. iT - 3.48e3T^{2}
61 1+74T+3.72e3T2 1 + 74T + 3.72e3T^{2}
67 1+96.9T+4.48e3T2 1 + 96.9T + 4.48e3T^{2}
71 1−6.92iT−5.04e3T2 1 - 6.92iT - 5.04e3T^{2}
73 1−132iT−5.32e3T2 1 - 132iT - 5.32e3T^{2}
79 1−103.iT−6.24e3T2 1 - 103. iT - 6.24e3T^{2}
83 1−117.T+6.88e3T2 1 - 117.T + 6.88e3T^{2}
89 1−14T+7.92e3T2 1 - 14T + 7.92e3T^{2}
97 1+24iT−9.40e3T2 1 + 24iT - 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

−10.65937747791654854796476766206, −9.934651375807434083794516905082, −8.549536670829549629125088396510, −8.097956461286962583103898298221, −7.17423019244942965620876426103, −6.21642031527777616489226316335, −5.17510636622496800536972186639, −3.96648652867301586078960601706, −3.18838228490656428972233363266, −1.65088139158424637742698326513, 0.19628945531262993470294254129, 1.74212986364623220156922886584, 3.24086455862927495421069637014, 4.49187887872670521108278987146, 4.98638146677466146881120208418, 6.33619822856741552008586386142, 7.44880672766954705033482671495, 7.973049587284638204204255670356, 9.056837631158755050813217789726, 9.606396380395079037075115522952

Graph of the ZZ-function along the critical line