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Properties

Label 2-1200-1.1-c3-0-15
Degree 22
Conductor 12001200
Sign 11
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4.43·7-s + 9·9-s + 3.43·11-s + 78.7·13-s + 53.1·17-s − 20.4·19-s − 13.3·21-s − 118.·23-s − 27·27-s + 168.·29-s + 61.0·31-s − 10.3·33-s − 246.·37-s − 236.·39-s + 422.·41-s − 362.·43-s + 170.·47-s − 323.·49-s − 159.·51-s − 546.·53-s + 61.3·57-s + 216.·59-s + 130.·61-s + 39.9·63-s + 614.·67-s + 354.·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.239·7-s + 0.333·9-s + 0.0941·11-s + 1.67·13-s + 0.758·17-s − 0.246·19-s − 0.138·21-s − 1.07·23-s − 0.192·27-s + 1.07·29-s + 0.353·31-s − 0.0543·33-s − 1.09·37-s − 0.969·39-s + 1.60·41-s − 1.28·43-s + 0.529·47-s − 0.942·49-s − 0.438·51-s − 1.41·53-s + 0.142·57-s + 0.478·59-s + 0.274·61-s + 0.0798·63-s + 1.12·67-s + 0.619·69-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: no
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 1.9406799701.940679970
L(12)L(\frac12) ≈\approx 1.9406799701.940679970
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+3T 1 + 3T
5 1 1
good7 1−4.43T+343T2 1 - 4.43T + 343T^{2}
11 1−3.43T+1.33e3T2 1 - 3.43T + 1.33e3T^{2}
13 1−78.7T+2.19e3T2 1 - 78.7T + 2.19e3T^{2}
17 1−53.1T+4.91e3T2 1 - 53.1T + 4.91e3T^{2}
19 1+20.4T+6.85e3T2 1 + 20.4T + 6.85e3T^{2}
23 1+118.T+1.21e4T2 1 + 118.T + 1.21e4T^{2}
29 1−168.T+2.43e4T2 1 - 168.T + 2.43e4T^{2}
31 1−61.0T+2.97e4T2 1 - 61.0T + 2.97e4T^{2}
37 1+246.T+5.06e4T2 1 + 246.T + 5.06e4T^{2}
41 1−422.T+6.89e4T2 1 - 422.T + 6.89e4T^{2}
43 1+362.T+7.95e4T2 1 + 362.T + 7.95e4T^{2}
47 1−170.T+1.03e5T2 1 - 170.T + 1.03e5T^{2}
53 1+546.T+1.48e5T2 1 + 546.T + 1.48e5T^{2}
59 1−216.T+2.05e5T2 1 - 216.T + 2.05e5T^{2}
61 1−130.T+2.26e5T2 1 - 130.T + 2.26e5T^{2}
67 1−614.T+3.00e5T2 1 - 614.T + 3.00e5T^{2}
71 1+324.T+3.57e5T2 1 + 324.T + 3.57e5T^{2}
73 1+88.8T+3.89e5T2 1 + 88.8T + 3.89e5T^{2}
79 1−1.13e3T+4.93e5T2 1 - 1.13e3T + 4.93e5T^{2}
83 1−758.T+5.71e5T2 1 - 758.T + 5.71e5T^{2}
89 1−195.T+7.04e5T2 1 - 195.T + 7.04e5T^{2}
97 1−521T+9.12e5T2 1 - 521T + 9.12e5T^{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

−9.436668028118038519227309944081, −8.417040653075729077707211503908, −7.88771010992851787689019347176, −6.63703279491758345981701194380, −6.10321999594169619831863839845, −5.19955176561685212083681972357, −4.17724365826440078712774198803, −3.30053302366629128402217145378, −1.78380369826522596803033591672, −0.77105254202752678220666731640, 0.77105254202752678220666731640, 1.78380369826522596803033591672, 3.30053302366629128402217145378, 4.17724365826440078712774198803, 5.19955176561685212083681972357, 6.10321999594169619831863839845, 6.63703279491758345981701194380, 7.88771010992851787689019347176, 8.417040653075729077707211503908, 9.436668028118038519227309944081

Graph of the ZZ-function along the critical line