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Properties

Label 2-1872-1.1-c3-0-56
Degree 22
Conductor 18721872
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 20·7-s + 24·11-s + 13·13-s + 30·17-s + 16·19-s − 72·23-s − 89·25-s + 282·29-s − 164·31-s + 120·35-s + 110·37-s + 126·41-s − 164·43-s − 204·47-s + 57·49-s + 738·53-s − 144·55-s + 120·59-s + 614·61-s − 78·65-s − 848·67-s + 132·71-s + 218·73-s − 480·77-s + 1.09e3·79-s + 552·83-s + ⋯
L(s)  = 1  − 0.536·5-s − 1.07·7-s + 0.657·11-s + 0.277·13-s + 0.428·17-s + 0.193·19-s − 0.652·23-s − 0.711·25-s + 1.80·29-s − 0.950·31-s + 0.579·35-s + 0.488·37-s + 0.479·41-s − 0.581·43-s − 0.633·47-s + 0.166·49-s + 1.91·53-s − 0.353·55-s + 0.264·59-s + 1.28·61-s − 0.148·65-s − 1.54·67-s + 0.220·71-s + 0.349·73-s − 0.710·77-s + 1.56·79-s + 0.729·83-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: −1-1
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: 11
Selberg data: (2, 1872, ( :3/2), −1)(2,\ 1872,\ (\ :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
13 1−pT 1 - p T
good5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
7 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
31 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
37 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
41 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
43 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
47 1+204T+p3T2 1 + 204 T + p^{3} T^{2}
53 1−738T+p3T2 1 - 738 T + p^{3} T^{2}
59 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
61 1−614T+p3T2 1 - 614 T + p^{3} T^{2}
67 1+848T+p3T2 1 + 848 T + p^{3} T^{2}
71 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
73 1−218T+p3T2 1 - 218 T + p^{3} T^{2}
79 1−1096T+p3T2 1 - 1096 T + p^{3} T^{2}
83 1−552T+p3T2 1 - 552 T + p^{3} T^{2}
89 1+210T+p3T2 1 + 210 T + p^{3} T^{2}
97 1+1726T+p3T2 1 + 1726 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

−8.460927541068129522516634245310, −7.72767882584697136002012332561, −6.79043842710023797969185955056, −6.23989418328161810465906319080, −5.29851559016372706222394964015, −4.08389471868059630471562959481, −3.56478878686693924459734333556, −2.53297029940447166104542452433, −1.13002115081879362538873988443, 0, 1.13002115081879362538873988443, 2.53297029940447166104542452433, 3.56478878686693924459734333556, 4.08389471868059630471562959481, 5.29851559016372706222394964015, 6.23989418328161810465906319080, 6.79043842710023797969185955056, 7.72767882584697136002012332561, 8.460927541068129522516634245310

Graph of the ZZ-function along the critical line