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Properties

Label 2-336-1.1-c5-0-18
Degree 22
Conductor 336336
Sign −1-1
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 64·5-s − 49·7-s + 81·9-s + 54·11-s + 738·13-s + 576·15-s − 848·17-s + 1.60e3·19-s + 441·21-s + 3.67e3·23-s + 971·25-s − 729·27-s − 4.33e3·29-s + 4.76e3·31-s − 486·33-s + 3.13e3·35-s − 2.09e3·37-s − 6.64e3·39-s − 6.11e3·41-s − 7.91e3·43-s − 5.18e3·45-s − 6.57e3·47-s + 2.40e3·49-s + 7.63e3·51-s − 7.89e3·53-s − 3.45e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.14·5-s − 0.377·7-s + 1/3·9-s + 0.134·11-s + 1.21·13-s + 0.660·15-s − 0.711·17-s + 1.01·19-s + 0.218·21-s + 1.44·23-s + 0.310·25-s − 0.192·27-s − 0.956·29-s + 0.889·31-s − 0.0776·33-s + 0.432·35-s − 0.251·37-s − 0.699·39-s − 0.568·41-s − 0.652·43-s − 0.381·45-s − 0.433·47-s + 1/7·49-s + 0.410·51-s − 0.386·53-s − 0.154·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 336, ( :5/2), −1)(2,\ 336,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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+p2T 1 + p^{2} T
7 1+p2T 1 + p^{2} T
good5 1+64T+p5T2 1 + 64 T + p^{5} T^{2}
11 1−54T+p5T2 1 - 54 T + p^{5} T^{2}
13 1−738T+p5T2 1 - 738 T + p^{5} T^{2}
17 1+848T+p5T2 1 + 848 T + p^{5} T^{2}
19 1−1604T+p5T2 1 - 1604 T + p^{5} T^{2}
23 1−3670T+p5T2 1 - 3670 T + p^{5} T^{2}
29 1+4330T+p5T2 1 + 4330 T + p^{5} T^{2}
31 1−4760T+p5T2 1 - 4760 T + p^{5} T^{2}
37 1+2094T+p5T2 1 + 2094 T + p^{5} T^{2}
41 1+6116T+p5T2 1 + 6116 T + p^{5} T^{2}
43 1+7916T+p5T2 1 + 7916 T + p^{5} T^{2}
47 1+6572T+p5T2 1 + 6572 T + p^{5} T^{2}
53 1+7894T+p5T2 1 + 7894 T + p^{5} T^{2}
59 1−41664T+p5T2 1 - 41664 T + p^{5} T^{2}
61 1+26570T+p5T2 1 + 26570 T + p^{5} T^{2}
67 1−41736T+p5T2 1 - 41736 T + p^{5} T^{2}
71 1+83574T+p5T2 1 + 83574 T + p^{5} T^{2}
73 1+42314T+p5T2 1 + 42314 T + p^{5} T^{2}
79 1+508T+p5T2 1 + 508 T + p^{5} T^{2}
83 1−8364T+p5T2 1 - 8364 T + p^{5} T^{2}
89 1+49220T+p5T2 1 + 49220 T + p^{5} T^{2}
97 1−159670T+p5T2 1 - 159670 T + p^{5} 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

−10.48159752562243141894724077633, −9.273550319399243687640192999938, −8.348757882836075576563077428416, −7.28417974855921185099907192910, −6.46563615155517240501636871886, −5.25954415483637372716836842909, −4.10053774400442202940911963221, −3.18187401842698923519102056507, −1.21109946677285471860517835713, 0, 1.21109946677285471860517835713, 3.18187401842698923519102056507, 4.10053774400442202940911963221, 5.25954415483637372716836842909, 6.46563615155517240501636871886, 7.28417974855921185099907192910, 8.348757882836075576563077428416, 9.273550319399243687640192999938, 10.48159752562243141894724077633

Graph of the ZZ-function along the critical line