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Properties

Label 2-9702-1.1-c1-0-109
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 − 8-s + 11-s − 2·13-s + 16-s − 6·17-s − 2·19-s − 22-s − 5·25-s + 2·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s + 2·37-s + 2·38-s + 6·41-s − 10·43-s + 44-s + 12·47-s + 5·50-s − 2·52-s + 12·53-s − 6·58-s + 12·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 1.52·43-s + 0.150·44-s + 1.75·47-s + 0.707·50-s − 0.277·52-s + 1.64·53-s − 0.787·58-s + 1.56·59-s + 1.28·61-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+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + 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 1−6T+pT2 1 - 6 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+2T+pT2 1 + 2 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

−7.22256681200124913942068687753, −6.86319691613869780218188863622, −6.13488512587456810544110169825, −5.40474200258705758947525517831, −4.41604514672277597522072986408, −3.93582844976437782803839905400, −2.64565898313481655439998497825, −2.25338768348159295239588485846, −1.09696957382779221676747338783, 0, 1.09696957382779221676747338783, 2.25338768348159295239588485846, 2.64565898313481655439998497825, 3.93582844976437782803839905400, 4.41604514672277597522072986408, 5.40474200258705758947525517831, 6.13488512587456810544110169825, 6.86319691613869780218188863622, 7.22256681200124913942068687753

Graph of the ZZ-function along the critical line