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Properties

Label 2-720-20.7-c1-0-12
Degree 22
Conductor 720720
Sign −0.525+0.850i-0.525 + 0.850i
Analytic cond. 5.749225.74922
Root an. cond. 2.397752.39775
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  + (−1 − 2i)5-s + (1 − i)13-s + (−5 − 5i)17-s + (−3 + 4i)25-s − 10i·29-s + (−7 − 7i)37-s − 10·41-s + 7i·49-s + (5 − 5i)53-s + 12·61-s + (−3 − i)65-s + (11 − 11i)73-s + (−5 + 15i)85-s + 10i·89-s + (−13 − 13i)97-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + (0.277 − 0.277i)13-s + (−1.21 − 1.21i)17-s + (−0.600 + 0.800i)25-s − 1.85i·29-s + (−1.15 − 1.15i)37-s − 1.56·41-s + i·49-s + (0.686 − 0.686i)53-s + 1.53·61-s + (−0.372 − 0.124i)65-s + (1.28 − 1.28i)73-s + (−0.542 + 1.62i)85-s + 1.05i·89-s + (−1.31 − 1.31i)97-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.525+0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((−0.525+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.525+0.850i-0.525 + 0.850i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(127,⋅)\chi_{720} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), −0.525+0.850i)(2,\ 720,\ (\ :1/2),\ -0.525 + 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.458065−0.821587i0.458065 - 0.821587i
L(12)L(\frac12) ≈\approx 0.458065−0.821587i0.458065 - 0.821587i
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 1
5 1+(1+2i)T 1 + (1 + 2i)T
good7 1−7iT2 1 - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1+(−1+i)T−13iT2 1 + (-1 + i)T - 13iT^{2}
17 1+(5+5i)T+17iT2 1 + (5 + 5i)T + 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1+23iT2 1 + 23iT^{2}
29 1+10iT−29T2 1 + 10iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(7+7i)T+37iT2 1 + (7 + 7i)T + 37iT^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+43iT2 1 + 43iT^{2}
47 1−47iT2 1 - 47iT^{2}
53 1+(−5+5i)T−53iT2 1 + (-5 + 5i)T - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1−12T+61T2 1 - 12T + 61T^{2}
67 1−67iT2 1 - 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−11+11i)T−73iT2 1 + (-11 + 11i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1−10iT−89T2 1 - 10iT - 89T^{2}
97 1+(13+13i)T+97iT2 1 + (13 + 13i)T + 97iT^{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

−10.01494075156774709424674860623, −9.155922408831269568252927866396, −8.480562004187249511904055025426, −7.60198276477324176105330257627, −6.65049688413170539207160463909, −5.46976334895499297820517657150, −4.63988777221551882064743399918, −3.67102638766190293415429729947, −2.18720762862321371885776946629, −0.47540063476749427317373765702, 1.86923672087254418009846353794, 3.23630675826133321837138035102, 4.09557281354080656170450891442, 5.31758959749386313518572698943, 6.63553547403116105017187727342, 6.91800161053221953059438822094, 8.256940296972529450224820556715, 8.780335037974028390004830251118, 10.08844439675758383541538758720, 10.69070048906599955411700646339

Graph of the ZZ-function along the critical line