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Properties

Label 2-320892-1.1-c1-0-32
Degree 22
Conductor 320892320892
Sign −1-1
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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 + 4·5-s + 7-s + 9-s + 13-s + 4·15-s − 17-s + 2·19-s + 21-s + 4·23-s + 11·25-s + 27-s − 29-s − 11·31-s + 4·35-s + 3·37-s + 39-s − 6·41-s − 3·43-s + 4·45-s − 9·47-s − 6·49-s − 51-s − 2·53-s + 2·57-s − 9·59-s + 12·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s + 1.03·15-s − 0.242·17-s + 0.458·19-s + 0.218·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 0.185·29-s − 1.97·31-s + 0.676·35-s + 0.493·37-s + 0.160·39-s − 0.937·41-s − 0.457·43-s + 0.596·45-s − 1.31·47-s − 6/7·49-s − 0.140·51-s − 0.274·53-s + 0.264·57-s − 1.17·59-s + 1.53·61-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: −1-1
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 320892, ( :1/2), −1)(2,\ 320892,\ (\ :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
11 1 1
13 1−T 1 - T
17 1+T 1 + T
good5 1−4T+pT2 1 - 4 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1+11T+pT2 1 + 11 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+3T+pT2 1 + 3 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+5T+pT2 1 + 5 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1−12T+pT2 1 - 12 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

−12.96708828043227, −12.76355934175619, −11.92703255917349, −11.33605048575227, −11.05167253307279, −10.43271226490260, −10.06263278287038, −9.609436871475098, −9.219014638524730, −8.782540384012605, −8.496592168175742, −7.660943956630998, −7.377432855726463, −6.699616684888043, −6.327559934422580, −5.849069214829047, −5.150653742455150, −5.050961913417431, −4.389397581192622, −3.411927156011473, −3.309541176811546, −2.491806010802484, −2.005916396555064, −1.549134770719489, −1.130767234639261, 0, 1.130767234639261, 1.549134770719489, 2.005916396555064, 2.491806010802484, 3.309541176811546, 3.411927156011473, 4.389397581192622, 5.050961913417431, 5.150653742455150, 5.849069214829047, 6.327559934422580, 6.699616684888043, 7.377432855726463, 7.660943956630998, 8.496592168175742, 8.782540384012605, 9.219014638524730, 9.609436871475098, 10.06263278287038, 10.43271226490260, 11.05167253307279, 11.33605048575227, 11.92703255917349, 12.76355934175619, 12.96708828043227

Graph of the ZZ-function along the critical line