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Properties

Label 2-720-20.3-c1-0-9
Degree 22
Conductor 720720
Sign −0.0898+0.995i-0.0898 + 0.995i
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  + (−2 + i)5-s + (−5 − 5i)13-s + (5 − 5i)17-s + (3 − 4i)25-s − 4i·29-s + (5 − 5i)37-s − 8·41-s − 7i·49-s + (−5 − 5i)53-s − 12·61-s + (15 + 5i)65-s + (5 + 5i)73-s + (−5 + 15i)85-s + 16i·89-s + (5 − 5i)97-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + (−1.38 − 1.38i)13-s + (1.21 − 1.21i)17-s + (0.600 − 0.800i)25-s − 0.742i·29-s + (0.821 − 0.821i)37-s − 1.24·41-s − i·49-s + (−0.686 − 0.686i)53-s − 1.53·61-s + (1.86 + 0.620i)65-s + (0.585 + 0.585i)73-s + (−0.542 + 1.62i)85-s + 1.69i·89-s + (0.507 − 0.507i)97-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.0898+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((−0.0898+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.0898+0.995i-0.0898 + 0.995i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(703,⋅)\chi_{720} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), −0.0898+0.995i)(2,\ 720,\ (\ :1/2),\ -0.0898 + 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.558495−0.611121i0.558495 - 0.611121i
L(12)L(\frac12) ≈\approx 0.558495−0.611121i0.558495 - 0.611121i
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+(2−i)T 1 + (2 - i)T
good7 1+7iT2 1 + 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1+(5+5i)T+13iT2 1 + (5 + 5i)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+4iT−29T2 1 + 4iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(−5+5i)T−37iT2 1 + (-5 + 5i)T - 37iT^{2}
41 1+8T+41T2 1 + 8T + 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+(−5−5i)T+73iT2 1 + (-5 - 5i)T + 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1−83iT2 1 - 83iT^{2}
89 1−16iT−89T2 1 - 16iT - 89T^{2}
97 1+(−5+5i)T−97iT2 1 + (-5 + 5i)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

−10.13501475744345699642418941512, −9.541155496830400545204402493216, −8.147513325142824767473202160957, −7.65505867975109223610531894634, −6.90350327093729244211903579518, −5.56676128846362329260603883788, −4.75168108136968396186086502134, −3.44409810353912618084281220687, −2.63212306855336792251199475316, −0.43842981696271729148507804072, 1.56151718046754628350246714915, 3.17301110087739539798135539875, 4.28295712761428366738200999132, 5.01419410160804988396575696349, 6.28591592061112047681363820206, 7.31861043615295832735158602451, 7.968249838097597296014444644572, 8.921850869445639689069105511047, 9.715288814539549451072271140271, 10.64065210830210825438694943800

Graph of the ZZ-function along the critical line