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Properties

Label 2-1800-1.1-c3-0-57
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  + 5·7-s − 14·11-s + 13-s − 46·17-s + 19·19-s + 46·23-s − 14·29-s + 133·31-s + 258·37-s − 84·41-s − 167·43-s − 410·47-s − 318·49-s − 456·53-s + 194·59-s − 17·61-s + 653·67-s − 828·71-s + 570·73-s − 70·77-s − 552·79-s − 142·83-s + 1.10e3·89-s + 5·91-s + 841·97-s − 552·101-s − 308·103-s + ⋯
L(s)  = 1  + 0.269·7-s − 0.383·11-s + 0.0213·13-s − 0.656·17-s + 0.229·19-s + 0.417·23-s − 0.0896·29-s + 0.770·31-s + 1.14·37-s − 0.319·41-s − 0.592·43-s − 1.27·47-s − 0.927·49-s − 1.18·53-s + 0.428·59-s − 0.0356·61-s + 1.19·67-s − 1.38·71-s + 0.913·73-s − 0.103·77-s − 0.786·79-s − 0.187·83-s + 1.31·89-s + 0.00575·91-s + 0.880·97-s − 0.543·101-s − 0.294·103-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 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
11 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
13 1−T+p3T2 1 - T + p^{3} T^{2}
17 1+46T+p3T2 1 + 46 T + p^{3} T^{2}
19 1−pT+p3T2 1 - p T + p^{3} T^{2}
23 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
29 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
31 1−133T+p3T2 1 - 133 T + p^{3} T^{2}
37 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
41 1+84T+p3T2 1 + 84 T + p^{3} T^{2}
43 1+167T+p3T2 1 + 167 T + p^{3} T^{2}
47 1+410T+p3T2 1 + 410 T + p^{3} T^{2}
53 1+456T+p3T2 1 + 456 T + p^{3} T^{2}
59 1−194T+p3T2 1 - 194 T + p^{3} T^{2}
61 1+17T+p3T2 1 + 17 T + p^{3} T^{2}
67 1−653T+p3T2 1 - 653 T + p^{3} T^{2}
71 1+828T+p3T2 1 + 828 T + p^{3} T^{2}
73 1−570T+p3T2 1 - 570 T + p^{3} T^{2}
79 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
83 1+142T+p3T2 1 + 142 T + p^{3} T^{2}
89 1−1104T+p3T2 1 - 1104 T + p^{3} T^{2}
97 1−841T+p3T2 1 - 841 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

−8.394062333003324819342546269676, −7.87622047089160165074485813137, −6.88344373330780905111651041619, −6.19406873968322098397404755407, −5.13901083687427881919391142961, −4.51277388064621196337264088417, −3.37899427814501525479819068000, −2.43147235528511051353944144449, −1.30291284351062520066448049432, 0, 1.30291284351062520066448049432, 2.43147235528511051353944144449, 3.37899427814501525479819068000, 4.51277388064621196337264088417, 5.13901083687427881919391142961, 6.19406873968322098397404755407, 6.88344373330780905111651041619, 7.87622047089160165074485813137, 8.394062333003324819342546269676

Graph of the ZZ-function along the critical line