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Properties

Label 2-9200-1.1-c1-0-146
Degree 22
Conductor 92009200
Sign −1-1
Analytic cond. 73.462373.4623
Root an. cond. 8.571018.57101
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  − 7-s − 3·9-s + 11-s + 13-s + 5·19-s − 23-s − 5·29-s + 2·31-s − 4·37-s − 5·41-s + 9·43-s + 6·47-s − 6·49-s + 2·53-s − 8·59-s − 8·61-s + 3·63-s − 8·67-s + 10·71-s − 3·73-s − 77-s + 3·79-s + 9·81-s − 3·83-s + 10·89-s − 91-s − 2·97-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 0.301·11-s + 0.277·13-s + 1.14·19-s − 0.208·23-s − 0.928·29-s + 0.359·31-s − 0.657·37-s − 0.780·41-s + 1.37·43-s + 0.875·47-s − 6/7·49-s + 0.274·53-s − 1.04·59-s − 1.02·61-s + 0.377·63-s − 0.977·67-s + 1.18·71-s − 0.351·73-s − 0.113·77-s + 0.337·79-s + 81-s − 0.329·83-s + 1.05·89-s − 0.104·91-s − 0.203·97-s + ⋯

Functional equation

Λ(s)=(9200s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9200s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92009200    =    24⋅52⋅232^{4} \cdot 5^{2} \cdot 23
Sign: −1-1
Analytic conductor: 73.462373.4623
Root analytic conductor: 8.571018.57101
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9200, ( :1/2), −1)(2,\ 9200,\ (\ :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
5 1 1
23 1+T 1 + T
good3 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1−9T+pT2 1 - 9 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+3T+pT2 1 + 3 T + p T^{2}
79 1−3T+pT2 1 - 3 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1−10T+pT2 1 - 10 T + 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

−7.47876576093360472676670536725, −6.61942238595646901909970429403, −5.97125616564562654158881484661, −5.44580395342077403040036316979, −4.63593629688090053857005431353, −3.64515207426874340591598364572, −3.15860055911463056565728903625, −2.26278994405622971296924013012, −1.18013169987124158680840935437, 0, 1.18013169987124158680840935437, 2.26278994405622971296924013012, 3.15860055911463056565728903625, 3.64515207426874340591598364572, 4.63593629688090053857005431353, 5.44580395342077403040036316979, 5.97125616564562654158881484661, 6.61942238595646901909970429403, 7.47876576093360472676670536725

Graph of the ZZ-function along the critical line