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Properties

Label 2-1872-1.1-c1-0-8
Degree 22
Conductor 18721872
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s − 13-s − 2·17-s + 4·23-s − 5·25-s + 6·29-s + 4·31-s − 2·37-s − 4·43-s + 10·47-s − 7·49-s + 10·53-s − 6·59-s − 6·61-s + 12·67-s + 2·71-s + 6·73-s + 16·79-s + 6·83-s − 4·89-s + 14·97-s + 6·101-s + 8·103-s − 8·107-s − 6·109-s + 10·113-s + ⋯
L(s)  = 1  + 1.80·11-s − 0.277·13-s − 0.485·17-s + 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.609·43-s + 1.45·47-s − 49-s + 1.37·53-s − 0.781·59-s − 0.768·61-s + 1.46·67-s + 0.237·71-s + 0.702·73-s + 1.80·79-s + 0.658·83-s − 0.423·89-s + 1.42·97-s + 0.597·101-s + 0.788·103-s − 0.773·107-s − 0.574·109-s + 0.940·113-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: 11
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: 00
Selberg data: (2, 1872, ( :1/2), 1)(2,\ 1872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9174890871.917489087
L(12)L(\frac12) ≈\approx 1.9174890871.917489087
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+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+4T+pT2 1 + 4 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

−9.184294838662426330599174276496, −8.603027162415544040212315107506, −7.63584831683610361882108019767, −6.69087411967754787433668721392, −6.28943189296774299916363973955, −5.09555003610369654280744252248, −4.24486360092282550965285188483, −3.43299310554595416240944294936, −2.19080394870072878544535794019, −0.986325425668759395181242182793, 0.986325425668759395181242182793, 2.19080394870072878544535794019, 3.43299310554595416240944294936, 4.24486360092282550965285188483, 5.09555003610369654280744252248, 6.28943189296774299916363973955, 6.69087411967754787433668721392, 7.63584831683610361882108019767, 8.603027162415544040212315107506, 9.184294838662426330599174276496

Graph of the ZZ-function along the critical line