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Properties

Label 2-1200-1.1-c3-0-3
Degree 22
Conductor 12001200
Sign 11
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·7-s + 9·9-s − 72·11-s + 6·13-s − 38·17-s − 52·19-s − 12·21-s + 152·23-s − 27·27-s − 78·29-s − 120·31-s + 216·33-s + 150·37-s − 18·39-s + 362·41-s − 484·43-s + 280·47-s − 327·49-s + 114·51-s + 670·53-s + 156·57-s − 696·59-s + 222·61-s + 36·63-s − 4·67-s − 456·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.215·7-s + 1/3·9-s − 1.97·11-s + 0.128·13-s − 0.542·17-s − 0.627·19-s − 0.124·21-s + 1.37·23-s − 0.192·27-s − 0.499·29-s − 0.695·31-s + 1.13·33-s + 0.666·37-s − 0.0739·39-s + 1.37·41-s − 1.71·43-s + 0.868·47-s − 0.953·49-s + 0.313·51-s + 1.73·53-s + 0.362·57-s − 1.53·59-s + 0.465·61-s + 0.0719·63-s − 0.00729·67-s − 0.795·69-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1200, ( :3/2), 1)(2,\ 1200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0740261741.074026174
L(12)L(\frac12) ≈\approx 1.0740261741.074026174
L(52)L(\frac{5}{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+pT 1 + p T
5 1 1
good7 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
13 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
17 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
19 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
23 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
29 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
31 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
37 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
41 1−362T+p3T2 1 - 362 T + p^{3} T^{2}
43 1+484T+p3T2 1 + 484 T + p^{3} T^{2}
47 1−280T+p3T2 1 - 280 T + p^{3} T^{2}
53 1−670T+p3T2 1 - 670 T + p^{3} T^{2}
59 1+696T+p3T2 1 + 696 T + p^{3} T^{2}
61 1−222T+p3T2 1 - 222 T + p^{3} T^{2}
67 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
71 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
73 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
79 1−8pT+p3T2 1 - 8 p T + p^{3} T^{2}
83 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
89 1−994T+p3T2 1 - 994 T + p^{3} T^{2}
97 1+1634T+p3T2 1 + 1634 T + p^{3} 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

−9.413546668410605712924156276610, −8.497310512738676789586005233622, −7.67479124765257075715448512391, −6.94819411649591222504256383414, −5.86472365838579624169533227947, −5.16711049966913734828213781660, −4.40244113680151271480742637498, −3.04375495940361895483561746749, −2.03237067325083737730616570480, −0.52752301244668289932876258540, 0.52752301244668289932876258540, 2.03237067325083737730616570480, 3.04375495940361895483561746749, 4.40244113680151271480742637498, 5.16711049966913734828213781660, 5.86472365838579624169533227947, 6.94819411649591222504256383414, 7.67479124765257075715448512391, 8.497310512738676789586005233622, 9.413546668410605712924156276610

Graph of the ZZ-function along the critical line