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Properties

Label 2-378-1.1-c3-0-16
Degree 22
Conductor 378378
Sign −1-1
Analytic cond. 22.302722.3027
Root an. cond. 4.722574.72257
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 5-s − 7·7-s − 8·8-s − 2·10-s + 44·11-s − 66·13-s + 14·14-s + 16·16-s − 7·17-s − 4·19-s + 4·20-s − 88·22-s + 86·23-s − 124·25-s + 132·26-s − 28·28-s − 176·29-s + 162·31-s − 32·32-s + 14·34-s − 7·35-s − 199·37-s + 8·38-s − 8·40-s + 363·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.0894·5-s − 0.377·7-s − 0.353·8-s − 0.0632·10-s + 1.20·11-s − 1.40·13-s + 0.267·14-s + 1/4·16-s − 0.0998·17-s − 0.0482·19-s + 0.0447·20-s − 0.852·22-s + 0.779·23-s − 0.991·25-s + 0.995·26-s − 0.188·28-s − 1.12·29-s + 0.938·31-s − 0.176·32-s + 0.0706·34-s − 0.0338·35-s − 0.884·37-s + 0.0341·38-s − 0.0316·40-s + 1.38·41-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: −1-1
Analytic conductor: 22.302722.3027
Root analytic conductor: 4.722574.72257
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 378, ( :3/2), −1)(2,\ 378,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+pT 1 + p T
3 1 1
7 1+pT 1 + p T
good5 1−T+p3T2 1 - T + p^{3} T^{2}
11 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
13 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
17 1+7T+p3T2 1 + 7 T + p^{3} T^{2}
19 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
23 1−86T+p3T2 1 - 86 T + p^{3} T^{2}
29 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
31 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
37 1+199T+p3T2 1 + 199 T + p^{3} T^{2}
41 1−363T+p3T2 1 - 363 T + p^{3} T^{2}
43 1+451T+p3T2 1 + 451 T + p^{3} T^{2}
47 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
53 1+174T+p3T2 1 + 174 T + p^{3} T^{2}
59 1+587T+p3T2 1 + 587 T + p^{3} T^{2}
61 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
67 1+560T+p3T2 1 + 560 T + p^{3} T^{2}
71 1+532T+p3T2 1 + 532 T + p^{3} T^{2}
73 1+854T+p3T2 1 + 854 T + p^{3} T^{2}
79 1+747T+p3T2 1 + 747 T + p^{3} T^{2}
83 1+613T+p3T2 1 + 613 T + p^{3} T^{2}
89 1+1266T+p3T2 1 + 1266 T + p^{3} T^{2}
97 1−64T+p3T2 1 - 64 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

−10.23998494972030166532686649877, −9.533026112852872069233046025419, −8.852735676386272387503669708737, −7.60442949687864534788333956386, −6.85915885032037728999264576853, −5.83202112566545823659617074467, −4.43447861842606633627014415804, −3.03316104694181115699210780651, −1.64423259205725316536322945214, 0, 1.64423259205725316536322945214, 3.03316104694181115699210780651, 4.43447861842606633627014415804, 5.83202112566545823659617074467, 6.85915885032037728999264576853, 7.60442949687864534788333956386, 8.852735676386272387503669708737, 9.533026112852872069233046025419, 10.23998494972030166532686649877

Graph of the ZZ-function along the critical line