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Properties

Label 2-9408-1.1-c1-0-130
Degree 22
Conductor 94089408
Sign −1-1
Analytic cond. 75.123275.1232
Root an. cond. 8.667368.66736
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  + 3-s − 2·5-s + 9-s + 2·11-s + 13-s − 2·15-s + 19-s − 25-s + 27-s − 4·29-s − 9·31-s + 2·33-s − 3·37-s + 39-s + 10·41-s − 5·43-s − 2·45-s + 6·47-s − 12·53-s − 4·55-s + 57-s − 12·59-s + 10·61-s − 2·65-s + 5·67-s − 6·71-s + 3·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 0.229·19-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.61·31-s + 0.348·33-s − 0.493·37-s + 0.160·39-s + 1.56·41-s − 0.762·43-s − 0.298·45-s + 0.875·47-s − 1.64·53-s − 0.539·55-s + 0.132·57-s − 1.56·59-s + 1.28·61-s − 0.248·65-s + 0.610·67-s − 0.712·71-s + 0.351·73-s + ⋯

Functional equation

Λ(s)=(9408s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9408s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94089408    =    26⋅3⋅722^{6} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 75.123275.1232
Root analytic conductor: 8.667368.66736
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9408, ( :1/2), −1)(2,\ 9408,\ (\ :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 1
3 1−T 1 - T
7 1 1
good5 1+2T+pT2 1 + 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1−6T+pT2 1 - 6 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 1−5T+pT2 1 - 5 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−7.58183307195557815202854342196, −6.83816225784114151325296720078, −6.04873695495652424409089915829, −5.25829752501167341687747079033, −4.33520014774866337516677825582, −3.76817176104573041637374040110, −3.25325181654870950802440024349, −2.18110487084687669897456722275, −1.29454970090328798188294397361, 0, 1.29454970090328798188294397361, 2.18110487084687669897456722275, 3.25325181654870950802440024349, 3.76817176104573041637374040110, 4.33520014774866337516677825582, 5.25829752501167341687747079033, 6.04873695495652424409089915829, 6.83816225784114151325296720078, 7.58183307195557815202854342196

Graph of the ZZ-function along the critical line