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Properties

Label 2-1872-1.1-c3-0-81
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 + 4·7-s + 36·11-s + 13·13-s − 66·17-s − 56·19-s + 96·23-s − 89·25-s − 222·29-s − 260·31-s + 24·35-s − 106·37-s + 90·41-s − 44·43-s + 168·47-s − 327·49-s − 30·53-s + 216·55-s + 348·59-s − 346·61-s + 78·65-s + 256·67-s − 168·71-s − 814·73-s + 144·77-s − 200·79-s + 1.23e3·83-s + ⋯
L(s)  = 1  + 0.536·5-s + 0.215·7-s + 0.986·11-s + 0.277·13-s − 0.941·17-s − 0.676·19-s + 0.870·23-s − 0.711·25-s − 1.42·29-s − 1.50·31-s + 0.115·35-s − 0.470·37-s + 0.342·41-s − 0.156·43-s + 0.521·47-s − 0.953·49-s − 0.0777·53-s + 0.529·55-s + 0.767·59-s − 0.726·61-s + 0.148·65-s + 0.466·67-s − 0.280·71-s − 1.30·73-s + 0.213·77-s − 0.284·79-s + 1.63·83-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 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
17 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
19 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
23 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
29 1+222T+p3T2 1 + 222 T + p^{3} T^{2}
31 1+260T+p3T2 1 + 260 T + p^{3} T^{2}
37 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
41 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
43 1+44T+p3T2 1 + 44 T + p^{3} T^{2}
47 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
53 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
59 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
61 1+346T+p3T2 1 + 346 T + p^{3} T^{2}
67 1−256T+p3T2 1 - 256 T + p^{3} T^{2}
71 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
73 1+814T+p3T2 1 + 814 T + p^{3} T^{2}
79 1+200T+p3T2 1 + 200 T + p^{3} T^{2}
83 1−1236T+p3T2 1 - 1236 T + p^{3} T^{2}
89 1+318T+p3T2 1 + 318 T + p^{3} T^{2}
97 1+502T+p3T2 1 + 502 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

−8.760655461053525863548133306562, −7.61777654038330747002093782203, −6.84475166749525088433853387170, −6.10401743314583280430330898378, −5.31912031151596970082541397253, −4.28206935869725760439513685793, −3.52955734910080000120246756361, −2.19120821516584182377322767414, −1.46632696507118953976319481681, 0, 1.46632696507118953976319481681, 2.19120821516584182377322767414, 3.52955734910080000120246756361, 4.28206935869725760439513685793, 5.31912031151596970082541397253, 6.10401743314583280430330898378, 6.84475166749525088433853387170, 7.61777654038330747002093782203, 8.760655461053525863548133306562

Graph of the ZZ-function along the critical line