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Properties

Label 2-9200-1.1-c1-0-125
Degree 22
Conductor 92009200
Sign −1-1
Analytic cond. 73.462373.4623
Root an. cond. 8.571018.57101
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s + 1.56·7-s + 3.56·9-s + 2·11-s − 0.561·13-s − 5.56·17-s + 2·19-s − 4·21-s − 23-s − 1.43·27-s + 0.123·29-s + 8.12·31-s − 5.12·33-s + 3.56·37-s + 1.43·39-s − 4.12·41-s − 10.2·43-s + 3.68·47-s − 4.56·49-s + 14.2·51-s − 4.43·53-s − 5.12·57-s + 5.56·59-s − 9.12·61-s + 5.56·63-s − 11.5·67-s + 2.56·69-s + ⋯
L(s)  = 1  − 1.47·3-s + 0.590·7-s + 1.18·9-s + 0.603·11-s − 0.155·13-s − 1.34·17-s + 0.458·19-s − 0.872·21-s − 0.208·23-s − 0.276·27-s + 0.0228·29-s + 1.45·31-s − 0.891·33-s + 0.585·37-s + 0.230·39-s − 0.643·41-s − 1.56·43-s + 0.537·47-s − 0.651·49-s + 1.99·51-s − 0.609·53-s − 0.678·57-s + 0.724·59-s − 1.16·61-s + 0.700·63-s − 1.41·67-s + 0.308·69-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: no
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+2.56T+3T2 1 + 2.56T + 3T^{2}
7 1−1.56T+7T2 1 - 1.56T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+0.561T+13T2 1 + 0.561T + 13T^{2}
17 1+5.56T+17T2 1 + 5.56T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
29 1−0.123T+29T2 1 - 0.123T + 29T^{2}
31 1−8.12T+31T2 1 - 8.12T + 31T^{2}
37 1−3.56T+37T2 1 - 3.56T + 37T^{2}
41 1+4.12T+41T2 1 + 4.12T + 41T^{2}
43 1+10.2T+43T2 1 + 10.2T + 43T^{2}
47 1−3.68T+47T2 1 - 3.68T + 47T^{2}
53 1+4.43T+53T2 1 + 4.43T + 53T^{2}
59 1−5.56T+59T2 1 - 5.56T + 59T^{2}
61 1+9.12T+61T2 1 + 9.12T + 61T^{2}
67 1+11.5T+67T2 1 + 11.5T + 67T^{2}
71 1−5T+71T2 1 - 5T + 71T^{2}
73 1+3.43T+73T2 1 + 3.43T + 73T^{2}
79 1−9.12T+79T2 1 - 9.12T + 79T^{2}
83 1+4.68T+83T2 1 + 4.68T + 83T^{2}
89 1−8T+89T2 1 - 8T + 89T^{2}
97 1−3.12T+97T2 1 - 3.12T + 97T^{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

−7.16192065730929202213912097384, −6.47259830934224116917751372546, −6.18192597524370587940477169572, −5.22116037407795759947282835345, −4.72670475861424113589724985353, −4.19779893021655536674084439223, −3.06489936847121350526971444981, −1.93597908794572034982698454926, −1.07473623487783211915639187462, 0, 1.07473623487783211915639187462, 1.93597908794572034982698454926, 3.06489936847121350526971444981, 4.19779893021655536674084439223, 4.72670475861424113589724985353, 5.22116037407795759947282835345, 6.18192597524370587940477169572, 6.47259830934224116917751372546, 7.16192065730929202213912097384

Graph of the ZZ-function along the critical line