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Properties

Label 2-399e2-1.1-c1-0-4
Degree 22
Conductor 159201159201
Sign 11
Analytic cond. 1271.221271.22
Root an. cond. 35.654235.6542
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  − 2·4-s − 5·13-s + 4·16-s − 5·25-s + 4·31-s + 11·37-s − 13·43-s + 10·52-s + 13·61-s − 8·64-s − 16·67-s − 17·73-s + 17·79-s − 14·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s − 1.38·13-s + 16-s − 25-s + 0.718·31-s + 1.80·37-s − 1.98·43-s + 1.38·52-s + 1.66·61-s − 64-s − 1.95·67-s − 1.98·73-s + 1.91·79-s − 1.42·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(159201s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(159201s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 159201159201    =    32⋅72⋅1923^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 1271.221271.22
Root analytic conductor: 35.654235.6542
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 159201, ( :1/2), 1)(2,\ 159201,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.67927807830.6792780783
L(12)L(\frac12) ≈\approx 0.67927807830.6792780783
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)
bad3 1 1
7 1 1
19 1 1
good2 1+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + 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+pT2 1 + p T^{2}
43 1+13T+pT2 1 + 13 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+17T+pT2 1 + 17 T + p T^{2}
79 1−17T+pT2 1 - 17 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + 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

−13.21878592788528, −13.01256611296387, −12.30557931253132, −11.82757566383845, −11.64594163018305, −10.82177097661674, −10.16987273015425, −9.967090694850236, −9.457052413142937, −9.108312438570539, −8.362185664299306, −8.016194634291997, −7.607364858381222, −6.960947153058113, −6.422617411883077, −5.729363188684800, −5.361555618854692, −4.684480433798374, −4.411434838184895, −3.813849826721663, −3.098202235689186, −2.594847932338125, −1.853094838402834, −1.091910029114458, −0.2685603309754153, 0.2685603309754153, 1.091910029114458, 1.853094838402834, 2.594847932338125, 3.098202235689186, 3.813849826721663, 4.411434838184895, 4.684480433798374, 5.361555618854692, 5.729363188684800, 6.422617411883077, 6.960947153058113, 7.607364858381222, 8.016194634291997, 8.362185664299306, 9.108312438570539, 9.457052413142937, 9.967090694850236, 10.16987273015425, 10.82177097661674, 11.64594163018305, 11.82757566383845, 12.30557931253132, 13.01256611296387, 13.21878592788528

Graph of the ZZ-function along the critical line