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Properties

Label 2-9702-1.1-c1-0-138
Degree 22
Conductor 97029702
Sign −1-1
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
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  − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 11-s − 2·13-s + 16-s − 3·17-s − 2·19-s + 3·20-s − 22-s − 3·23-s + 4·25-s + 2·26-s − 2·31-s − 32-s + 3·34-s + 8·37-s + 2·38-s − 3·40-s − 9·41-s − 4·43-s + 44-s + 3·46-s + 3·47-s − 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.670·20-s − 0.213·22-s − 0.625·23-s + 4/5·25-s + 0.392·26-s − 0.359·31-s − 0.176·32-s + 0.514·34-s + 1.31·37-s + 0.324·38-s − 0.474·40-s − 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.442·46-s + 0.437·47-s − 0.565·50-s + ⋯

Functional equation

Λ(s)=(9702s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9702s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97029702    =    2⋅32⋅72⋅112 \cdot 3^{2} \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9702, ( :1/2), −1)(2,\ 9702,\ (\ :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+T 1 + T
3 1 1
7 1 1
11 1−T 1 - T
good5 1−3T+pT2 1 - 3 T + p T^{2}
13 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+3T+pT2 1 + 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+5T+pT2 1 + 5 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

−7.30172107412404859009345345116, −6.59616902877669987132874041325, −6.17527511155602092999588134751, −5.43046248487910812180376537136, −4.68780530333625811906142492584, −3.73894887268912114086430003538, −2.63573240115714471093680970623, −2.09425326723244478904619972866, −1.34095205027911786143302873504, 0, 1.34095205027911786143302873504, 2.09425326723244478904619972866, 2.63573240115714471093680970623, 3.73894887268912114086430003538, 4.68780530333625811906142492584, 5.43046248487910812180376537136, 6.17527511155602092999588134751, 6.59616902877669987132874041325, 7.30172107412404859009345345116

Graph of the ZZ-function along the critical line