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Properties

Label 2-720-5.4-c5-0-63
Degree 22
Conductor 720720
Sign −0.983+0.178i-0.983 + 0.178i
Analytic cond. 115.476115.476
Root an. cond. 10.745910.7459
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−55 + 10i)5-s − 158i·7-s − 148·11-s − 684i·13-s − 2.04e3i·17-s + 2.22e3·19-s + 1.24e3i·23-s + (2.92e3 − 1.10e3i)25-s − 270·29-s + 2.04e3·31-s + (1.58e3 + 8.69e3i)35-s − 4.37e3i·37-s + 2.39e3·41-s + 2.29e3i·43-s − 1.06e4i·47-s + ⋯
L(s)  = 1  + (−0.983 + 0.178i)5-s − 1.21i·7-s − 0.368·11-s − 1.12i·13-s − 1.71i·17-s + 1.41·19-s + 0.491i·23-s + (0.936 − 0.352i)25-s − 0.0596·29-s + 0.382·31-s + (0.218 + 1.19i)35-s − 0.525i·37-s + 0.222·41-s + 0.189i·43-s − 0.705i·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.983+0.178i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+5/2) L(s)=((−0.983+0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.983+0.178i-0.983 + 0.178i
Analytic conductor: 115.476115.476
Root analytic conductor: 10.745910.7459
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ720(289,⋅)\chi_{720} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :5/2), −0.983+0.178i)(2,\ 720,\ (\ :5/2),\ -0.983 + 0.178i)

Particular Values

L(3)L(3) ≈\approx 1.1234610281.123461028
L(12)L(\frac12) ≈\approx 1.1234610281.123461028
L(72)L(\frac{7}{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+(55−10i)T 1 + (55 - 10i)T
good7 1+158iT−1.68e4T2 1 + 158iT - 1.68e4T^{2}
11 1+148T+1.61e5T2 1 + 148T + 1.61e5T^{2}
13 1+684iT−3.71e5T2 1 + 684iT - 3.71e5T^{2}
17 1+2.04e3iT−1.41e6T2 1 + 2.04e3iT - 1.41e6T^{2}
19 1−2.22e3T+2.47e6T2 1 - 2.22e3T + 2.47e6T^{2}
23 1−1.24e3iT−6.43e6T2 1 - 1.24e3iT - 6.43e6T^{2}
29 1+270T+2.05e7T2 1 + 270T + 2.05e7T^{2}
31 1−2.04e3T+2.86e7T2 1 - 2.04e3T + 2.86e7T^{2}
37 1+4.37e3iT−6.93e7T2 1 + 4.37e3iT - 6.93e7T^{2}
41 1−2.39e3T+1.15e8T2 1 - 2.39e3T + 1.15e8T^{2}
43 1−2.29e3iT−1.47e8T2 1 - 2.29e3iT - 1.47e8T^{2}
47 1+1.06e4iT−2.29e8T2 1 + 1.06e4iT - 2.29e8T^{2}
53 1−2.96e3iT−4.18e8T2 1 - 2.96e3iT - 4.18e8T^{2}
59 1−3.97e4T+7.14e8T2 1 - 3.97e4T + 7.14e8T^{2}
61 1+4.22e4T+8.44e8T2 1 + 4.22e4T + 8.44e8T^{2}
67 1+3.20e4iT−1.35e9T2 1 + 3.20e4iT - 1.35e9T^{2}
71 1+4.24e3T+1.80e9T2 1 + 4.24e3T + 1.80e9T^{2}
73 1+3.01e4iT−2.07e9T2 1 + 3.01e4iT - 2.07e9T^{2}
79 1−3.52e4T+3.07e9T2 1 - 3.52e4T + 3.07e9T^{2}
83 1−2.78e4iT−3.93e9T2 1 - 2.78e4iT - 3.93e9T^{2}
89 1+8.52e4T+5.58e9T2 1 + 8.52e4T + 5.58e9T^{2}
97 1+9.72e4iT−8.58e9T2 1 + 9.72e4iT - 8.58e9T^{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

−9.355668036513258762201909828199, −8.077400479934443942902778531018, −7.47366335870589045784942961799, −7.00001335816922293298812854567, −5.48695229653567532681706656070, −4.65162063057916416811697579632, −3.55316429666574274638435600366, −2.86044218469418759283312630677, −0.940046955472601381627138948055, −0.30352029737504893863687685317, 1.28083330032161261943830772669, 2.50743321589217937955076141410, 3.62508818867320182130947131949, 4.59151064934415568329109083736, 5.58798640295237629456379136168, 6.52255924616614531058194645684, 7.58788739672034986071780418237, 8.423186770586250797940167920172, 8.985464609204727567999621514017, 9.997237747398641560083973834876

Graph of the ZZ-function along the critical line