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Properties

Label 2-1872-1.1-c3-0-46
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  + 7·5-s + 21·7-s + 6·11-s + 13·13-s + 115·17-s + 46·19-s + 144·23-s − 76·25-s + 162·29-s − 180·31-s + 147·35-s + 13·37-s − 192·41-s + 33·43-s + 383·47-s + 98·49-s − 288·53-s + 42·55-s + 442·59-s − 680·61-s + 91·65-s + 722·67-s − 207·71-s + 274·73-s + 126·77-s + 936·79-s − 1.20e3·83-s + ⋯
L(s)  = 1  + 0.626·5-s + 1.13·7-s + 0.164·11-s + 0.277·13-s + 1.64·17-s + 0.555·19-s + 1.30·23-s − 0.607·25-s + 1.03·29-s − 1.04·31-s + 0.709·35-s + 0.0577·37-s − 0.731·41-s + 0.117·43-s + 1.18·47-s + 2/7·49-s − 0.746·53-s + 0.102·55-s + 0.975·59-s − 1.42·61-s + 0.173·65-s + 1.31·67-s − 0.346·71-s + 0.439·73-s + 0.186·77-s + 1.33·79-s − 1.59·83-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 3.5575885313.557588531
L(12)L(\frac12) ≈\approx 3.5575885313.557588531
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 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
7 1−3pT+p3T2 1 - 3 p T + p^{3} T^{2}
11 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
17 1−115T+p3T2 1 - 115 T + p^{3} T^{2}
19 1−46T+p3T2 1 - 46 T + p^{3} T^{2}
23 1−144T+p3T2 1 - 144 T + p^{3} T^{2}
29 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
31 1+180T+p3T2 1 + 180 T + p^{3} T^{2}
37 1−13T+p3T2 1 - 13 T + p^{3} T^{2}
41 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
43 1−33T+p3T2 1 - 33 T + p^{3} T^{2}
47 1−383T+p3T2 1 - 383 T + p^{3} T^{2}
53 1+288T+p3T2 1 + 288 T + p^{3} T^{2}
59 1−442T+p3T2 1 - 442 T + p^{3} T^{2}
61 1+680T+p3T2 1 + 680 T + p^{3} T^{2}
67 1−722T+p3T2 1 - 722 T + p^{3} T^{2}
71 1+207T+p3T2 1 + 207 T + p^{3} T^{2}
73 1−274T+p3T2 1 - 274 T + p^{3} T^{2}
79 1−936T+p3T2 1 - 936 T + p^{3} T^{2}
83 1+1204T+p3T2 1 + 1204 T + p^{3} T^{2}
89 1−966T+p3T2 1 - 966 T + p^{3} T^{2}
97 1+138T+p3T2 1 + 138 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

−8.901826782215708878460649953225, −8.022095303096988216941686101327, −7.44832115774631302037837075595, −6.44976355644166394511906901157, −5.44649340739693040798376149835, −5.07466339228531419562784624795, −3.87479199434103494647863708531, −2.88498306112293453404444618372, −1.68702901703188181414439591414, −0.974353601692816649912859366023, 0.974353601692816649912859366023, 1.68702901703188181414439591414, 2.88498306112293453404444618372, 3.87479199434103494647863708531, 5.07466339228531419562784624795, 5.44649340739693040798376149835, 6.44976355644166394511906901157, 7.44832115774631302037837075595, 8.022095303096988216941686101327, 8.901826782215708878460649953225

Graph of the ZZ-function along the critical line