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Properties

Label 2-1872-1.1-c1-0-26
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 2·11-s − 13-s + 3·17-s − 6·19-s − 4·23-s − 4·25-s − 2·29-s − 4·31-s − 35-s + 3·37-s + 5·43-s + 13·47-s − 6·49-s − 12·53-s − 2·55-s − 10·59-s − 8·61-s − 65-s + 2·67-s − 5·71-s − 10·73-s + 2·77-s + 4·79-s + 3·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.493·37-s + 0.762·43-s + 1.89·47-s − 6/7·49-s − 1.64·53-s − 0.269·55-s − 1.30·59-s − 1.02·61-s − 0.124·65-s + 0.244·67-s − 0.593·71-s − 1.17·73-s + 0.227·77-s + 0.450·79-s + 0.325·85-s − 0.635·89-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :1/2), −1)(2,\ 1872,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−14T+pT2 1 - 14 T + p 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

−8.963928209525859216279788816156, −7.931818475756913627713932784539, −7.42085241738992224550819026024, −6.19695969612857050602479487947, −5.84229781714116440214520488100, −4.73920502052987077827176335423, −3.82010897415061793912288152862, −2.71625515373288996093576035666, −1.77354291627119599659363661895, 0, 1.77354291627119599659363661895, 2.71625515373288996093576035666, 3.82010897415061793912288152862, 4.73920502052987077827176335423, 5.84229781714116440214520488100, 6.19695969612857050602479487947, 7.42085241738992224550819026024, 7.931818475756913627713932784539, 8.963928209525859216279788816156

Graph of the ZZ-function along the critical line