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Properties

Label 2-1872-1.1-c1-0-4
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  + 5-s − 5·7-s − 2·11-s − 13-s + 3·17-s + 2·19-s + 4·23-s − 4·25-s + 6·29-s + 4·31-s − 5·35-s + 11·37-s − 8·41-s + 43-s + 9·47-s + 18·49-s + 12·53-s − 2·55-s + 6·59-s − 65-s − 6·67-s + 7·71-s − 2·73-s + 10·77-s − 12·79-s − 16·83-s + 3·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.88·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s + 1.11·29-s + 0.718·31-s − 0.845·35-s + 1.80·37-s − 1.24·41-s + 0.152·43-s + 1.31·47-s + 18/7·49-s + 1.64·53-s − 0.269·55-s + 0.781·59-s − 0.124·65-s − 0.733·67-s + 0.830·71-s − 0.234·73-s + 1.13·77-s − 1.35·79-s − 1.75·83-s + 0.325·85-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 1.3667164281.366716428
L(12)L(\frac12) ≈\approx 1.3667164281.366716428
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+5T+pT2 1 + 5 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 1−2T+pT2 1 - 2 T + 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 1−11T+pT2 1 - 11 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−9.399377966647754514275339054317, −8.571559742557125580918706419247, −7.51880508752900332556843831011, −6.82508995489029179736833518542, −6.01472830731285355227624329326, −5.41568943920328063726692378316, −4.19080409729406201912090076952, −3.11607344408057507029997990678, −2.55679172121781849822247168687, −0.77347272515720828453769161406, 0.77347272515720828453769161406, 2.55679172121781849822247168687, 3.11607344408057507029997990678, 4.19080409729406201912090076952, 5.41568943920328063726692378316, 6.01472830731285355227624329326, 6.82508995489029179736833518542, 7.51880508752900332556843831011, 8.571559742557125580918706419247, 9.399377966647754514275339054317

Graph of the ZZ-function along the critical line