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Properties

Label 2-9600-1.1-c1-0-110
Degree 22
Conductor 96009600
Sign −1-1
Analytic cond. 76.656376.6563
Root an. cond. 8.755368.75536
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  + 3-s − 3.12·7-s + 9-s − 2·11-s + 3.12·13-s − 7.12·17-s + 3.12·19-s − 3.12·21-s + 3.12·23-s + 27-s + 8.24·29-s − 1.12·31-s − 2·33-s − 3.12·37-s + 3.12·39-s − 2·41-s + 10.2·43-s − 4.87·47-s + 2.75·49-s − 7.12·51-s − 10·53-s + 3.12·57-s − 6·59-s − 2·61-s − 3.12·63-s + 10.2·67-s + 3.12·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.18·7-s + 0.333·9-s − 0.603·11-s + 0.866·13-s − 1.72·17-s + 0.716·19-s − 0.681·21-s + 0.651·23-s + 0.192·27-s + 1.53·29-s − 0.201·31-s − 0.348·33-s − 0.513·37-s + 0.500·39-s − 0.312·41-s + 1.56·43-s − 0.711·47-s + 0.393·49-s − 0.997·51-s − 1.37·53-s + 0.413·57-s − 0.781·59-s − 0.256·61-s − 0.393·63-s + 1.25·67-s + 0.375·69-s + ⋯

Functional equation

Λ(s)=(9600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 96009600    =    27⋅3⋅522^{7} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 76.656376.6563
Root analytic conductor: 8.755368.75536
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9600, ( :1/2), −1)(2,\ 9600,\ (\ :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
5 1 1
good7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−3.12T+13T2 1 - 3.12T + 13T^{2}
17 1+7.12T+17T2 1 + 7.12T + 17T^{2}
19 1−3.12T+19T2 1 - 3.12T + 19T^{2}
23 1−3.12T+23T2 1 - 3.12T + 23T^{2}
29 1−8.24T+29T2 1 - 8.24T + 29T^{2}
31 1+1.12T+31T2 1 + 1.12T + 31T^{2}
37 1+3.12T+37T2 1 + 3.12T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−10.2T+43T2 1 - 10.2T + 43T^{2}
47 1+4.87T+47T2 1 + 4.87T + 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−10.2T+67T2 1 - 10.2T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+12.2T+73T2 1 + 12.2T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+10T+97T2 1 + 10T + 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

−7.29258196287385806982992246008, −6.58937695543102705965759717968, −6.26306015914245922683108662934, −5.22482614286641376420958949185, −4.49932585098856861215033379606, −3.67054016579666201725284017080, −2.99423792522344929367161348330, −2.41752823443718083830712036522, −1.23548665275784589508844217394, 0, 1.23548665275784589508844217394, 2.41752823443718083830712036522, 2.99423792522344929367161348330, 3.67054016579666201725284017080, 4.49932585098856861215033379606, 5.22482614286641376420958949185, 6.26306015914245922683108662934, 6.58937695543102705965759717968, 7.29258196287385806982992246008

Graph of the ZZ-function along the critical line