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Properties

Label 2-1800-1.1-c3-0-39
Degree 22
Conductor 18001800
Sign −1-1
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 34·7-s + 18·11-s − 12·13-s + 106·17-s − 44·19-s − 56·23-s + 270·29-s + 204·31-s − 120·37-s + 80·41-s − 536·43-s + 536·47-s + 813·49-s − 542·53-s − 174·59-s + 186·61-s − 332·67-s − 132·71-s + 602·73-s − 612·77-s − 548·79-s + 492·83-s − 1.05e3·89-s + 408·91-s − 482·97-s + 1.21e3·101-s − 898·103-s + ⋯
L(s)  = 1  − 1.83·7-s + 0.493·11-s − 0.256·13-s + 1.51·17-s − 0.531·19-s − 0.507·23-s + 1.72·29-s + 1.18·31-s − 0.533·37-s + 0.304·41-s − 1.90·43-s + 1.66·47-s + 2.37·49-s − 1.40·53-s − 0.383·59-s + 0.390·61-s − 0.605·67-s − 0.220·71-s + 0.965·73-s − 0.905·77-s − 0.780·79-s + 0.650·83-s − 1.25·89-s + 0.470·91-s − 0.504·97-s + 1.19·101-s − 0.859·103-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1800, ( :3/2), −1)(2,\ 1800,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
good7 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
11 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
13 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
17 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
19 1+44T+p3T2 1 + 44 T + p^{3} T^{2}
23 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
29 1−270T+p3T2 1 - 270 T + p^{3} T^{2}
31 1−204T+p3T2 1 - 204 T + p^{3} T^{2}
37 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
41 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
43 1+536T+p3T2 1 + 536 T + p^{3} T^{2}
47 1−536T+p3T2 1 - 536 T + p^{3} T^{2}
53 1+542T+p3T2 1 + 542 T + p^{3} T^{2}
59 1+174T+p3T2 1 + 174 T + p^{3} T^{2}
61 1−186T+p3T2 1 - 186 T + p^{3} T^{2}
67 1+332T+p3T2 1 + 332 T + p^{3} T^{2}
71 1+132T+p3T2 1 + 132 T + p^{3} T^{2}
73 1−602T+p3T2 1 - 602 T + p^{3} T^{2}
79 1+548T+p3T2 1 + 548 T + p^{3} T^{2}
83 1−492T+p3T2 1 - 492 T + p^{3} T^{2}
89 1+1052T+p3T2 1 + 1052 T + p^{3} T^{2}
97 1+482T+p3T2 1 + 482 T + p^{3} T^{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

−8.584232553168694314021033728585, −7.72876181082921071888120306898, −6.65113326413272609196858278755, −6.36475773943266278649277295625, −5.38424602421298264458701189354, −4.22572386700575919813598731174, −3.33768781813259207778091142683, −2.67389012941901964485779336576, −1.13503637543123197955099945814, 0, 1.13503637543123197955099945814, 2.67389012941901964485779336576, 3.33768781813259207778091142683, 4.22572386700575919813598731174, 5.38424602421298264458701189354, 6.36475773943266278649277295625, 6.65113326413272609196858278755, 7.72876181082921071888120306898, 8.584232553168694314021033728585

Graph of the ZZ-function along the critical line