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Properties

Label 2-9408-1.1-c1-0-157
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 + 2·15-s − 2·17-s − 2·23-s − 25-s + 27-s − 6·29-s − 4·31-s − 2·33-s − 6·37-s + 2·41-s + 2·45-s − 2·51-s + 6·53-s − 4·55-s − 12·59-s + 12·61-s − 12·67-s − 2·69-s + 10·71-s − 12·73-s − 75-s − 12·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.485·17-s − 0.417·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.312·41-s + 0.298·45-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 1.53·61-s − 1.46·67-s − 0.240·69-s + 1.18·71-s − 1.40·73-s − 0.115·75-s − 1.35·79-s + 1/9·81-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+pT2 1 + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+2T+pT2 1 + 2 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+6T+pT2 1 + 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+12T+pT2 1 + 12 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−7.38152233350264094134261020512, −6.77839586162877146619173499201, −5.82353022355995261190831502319, −5.49576071035680018496349915277, −4.53079591074299736281466223085, −3.78711488789932938503782185219, −2.91557507898183341940151924426, −2.14776468086628798905550068389, −1.54646372600141965125612720140, 0, 1.54646372600141965125612720140, 2.14776468086628798905550068389, 2.91557507898183341940151924426, 3.78711488789932938503782185219, 4.53079591074299736281466223085, 5.49576071035680018496349915277, 5.82353022355995261190831502319, 6.77839586162877146619173499201, 7.38152233350264094134261020512

Graph of the ZZ-function along the critical line