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Properties

Label 2-1800-1.1-c3-0-19
Degree 22
Conductor 18001800
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·7-s + 56·11-s + 86·13-s − 106·17-s + 4·19-s + 136·23-s + 206·29-s − 152·31-s − 282·37-s + 246·41-s − 412·43-s + 40·47-s + 57·49-s − 126·53-s − 56·59-s − 2·61-s + 388·67-s + 672·71-s − 1.17e3·73-s − 1.12e3·77-s + 408·79-s + 668·83-s − 66·89-s − 1.72e3·91-s + 926·97-s + 198·101-s + 1.53e3·103-s + ⋯
L(s)  = 1  − 1.07·7-s + 1.53·11-s + 1.83·13-s − 1.51·17-s + 0.0482·19-s + 1.23·23-s + 1.31·29-s − 0.880·31-s − 1.25·37-s + 0.937·41-s − 1.46·43-s + 0.124·47-s + 0.166·49-s − 0.326·53-s − 0.123·59-s − 0.00419·61-s + 0.707·67-s + 1.12·71-s − 1.87·73-s − 1.65·77-s + 0.581·79-s + 0.883·83-s − 0.0786·89-s − 1.98·91-s + 0.969·97-s + 0.195·101-s + 1.46·103-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: 11
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: 00
Selberg data: (2, 1800, ( :3/2), 1)(2,\ 1800,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.1797188832.179718883
L(12)L(\frac12) ≈\approx 2.1797188832.179718883
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+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
13 1−86T+p3T2 1 - 86 T + p^{3} T^{2}
17 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
19 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
23 1−136T+p3T2 1 - 136 T + p^{3} T^{2}
29 1−206T+p3T2 1 - 206 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
41 1−6pT+p3T2 1 - 6 p T + p^{3} T^{2}
43 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
47 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
53 1+126T+p3T2 1 + 126 T + p^{3} T^{2}
59 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
61 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
67 1−388T+p3T2 1 - 388 T + p^{3} T^{2}
71 1−672T+p3T2 1 - 672 T + p^{3} T^{2}
73 1+1170T+p3T2 1 + 1170 T + p^{3} T^{2}
79 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
83 1−668T+p3T2 1 - 668 T + p^{3} T^{2}
89 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
97 1−926T+p3T2 1 - 926 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

−8.925312413838315296153567844285, −8.455320185200340270240312583880, −6.95383968431445259905055162589, −6.59496316899241756784596515146, −5.98217654583354121269046679930, −4.69809004211638301673652274808, −3.75589351529804126117477206645, −3.18847020563996099663551912861, −1.74854948086973600117930304728, −0.71893256481158632138902772625, 0.71893256481158632138902772625, 1.74854948086973600117930304728, 3.18847020563996099663551912861, 3.75589351529804126117477206645, 4.69809004211638301673652274808, 5.98217654583354121269046679930, 6.59496316899241756784596515146, 6.95383968431445259905055162589, 8.455320185200340270240312583880, 8.925312413838315296153567844285

Graph of the ZZ-function along the critical line