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Properties

Label 2-1872-1.1-c3-0-23
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
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  + 16·5-s − 28·7-s + 34·11-s − 13·13-s − 138·17-s − 108·19-s − 52·23-s + 131·25-s + 190·29-s + 176·31-s − 448·35-s + 342·37-s − 240·41-s + 140·43-s + 454·47-s + 441·49-s − 198·53-s + 544·55-s − 154·59-s + 34·61-s − 208·65-s + 656·67-s + 550·71-s + 614·73-s − 952·77-s − 8·79-s + 762·83-s + ⋯
L(s)  = 1  + 1.43·5-s − 1.51·7-s + 0.931·11-s − 0.277·13-s − 1.96·17-s − 1.30·19-s − 0.471·23-s + 1.04·25-s + 1.21·29-s + 1.01·31-s − 2.16·35-s + 1.51·37-s − 0.914·41-s + 0.496·43-s + 1.40·47-s + 9/7·49-s − 0.513·53-s + 1.33·55-s − 0.339·59-s + 0.0713·61-s − 0.396·65-s + 1.19·67-s + 0.919·71-s + 0.984·73-s − 1.40·77-s − 0.0113·79-s + 1.00·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :3/2), 1)(2,\ 1872,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0901237322.090123732
L(12)L(\frac12) ≈\approx 2.0901237322.090123732
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
13 1+pT 1 + p T
good5 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
7 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
11 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
17 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
19 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
23 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
29 1−190T+p3T2 1 - 190 T + p^{3} T^{2}
31 1−176T+p3T2 1 - 176 T + p^{3} T^{2}
37 1−342T+p3T2 1 - 342 T + p^{3} T^{2}
41 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
43 1−140T+p3T2 1 - 140 T + p^{3} T^{2}
47 1−454T+p3T2 1 - 454 T + p^{3} T^{2}
53 1+198T+p3T2 1 + 198 T + p^{3} T^{2}
59 1+154T+p3T2 1 + 154 T + p^{3} T^{2}
61 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
67 1−656T+p3T2 1 - 656 T + p^{3} T^{2}
71 1−550T+p3T2 1 - 550 T + p^{3} T^{2}
73 1−614T+p3T2 1 - 614 T + p^{3} T^{2}
79 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
83 1−762T+p3T2 1 - 762 T + p^{3} T^{2}
89 1−444T+p3T2 1 - 444 T + p^{3} T^{2}
97 1−1022T+p3T2 1 - 1022 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

−9.153783204566902111820325221224, −8.329023603381918006652985013027, −6.82006329789906365421283936615, −6.39920708926293870666574683470, −6.09658346351003424337354943804, −4.74767910479822582247339848330, −3.93779674113940551225010265874, −2.64208343378806915709390516261, −2.10152691088045426321000263779, −0.64969236614616481216090046033, 0.64969236614616481216090046033, 2.10152691088045426321000263779, 2.64208343378806915709390516261, 3.93779674113940551225010265874, 4.74767910479822582247339848330, 6.09658346351003424337354943804, 6.39920708926293870666574683470, 6.82006329789906365421283936615, 8.329023603381918006652985013027, 9.153783204566902111820325221224

Graph of the ZZ-function along the critical line