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Properties

Label 2-336-1.1-c3-0-14
Degree 22
Conductor 336336
Sign −1-1
Analytic cond. 19.824619.8246
Root an. cond. 4.452484.45248
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 10.5·5-s − 7·7-s + 9·9-s − 34.7·11-s − 37.2·13-s − 31.6·15-s − 10.5·17-s + 58.5·19-s + 21·21-s + 125.·23-s − 13.7·25-s − 27·27-s − 35.4·29-s − 291.·31-s + 104.·33-s − 73.8·35-s − 259.·37-s + 111.·39-s − 338.·41-s − 6.80·43-s + 94.9·45-s − 250.·47-s + 49·49-s + 31.6·51-s − 536.·53-s − 366.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.943·5-s − 0.377·7-s + 0.333·9-s − 0.952·11-s − 0.795·13-s − 0.544·15-s − 0.150·17-s + 0.707·19-s + 0.218·21-s + 1.13·23-s − 0.109·25-s − 0.192·27-s − 0.226·29-s − 1.69·31-s + 0.549·33-s − 0.356·35-s − 1.15·37-s + 0.459·39-s − 1.28·41-s − 0.0241·43-s + 0.314·45-s − 0.778·47-s + 0.142·49-s + 0.0868·51-s − 1.39·53-s − 0.898·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 19.824619.8246
Root analytic conductor: 4.452484.45248
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 336, ( :3/2), −1)(2,\ 336,\ (\ :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 1
3 1+3T 1 + 3T
7 1+7T 1 + 7T
good5 1−10.5T+125T2 1 - 10.5T + 125T^{2}
11 1+34.7T+1.33e3T2 1 + 34.7T + 1.33e3T^{2}
13 1+37.2T+2.19e3T2 1 + 37.2T + 2.19e3T^{2}
17 1+10.5T+4.91e3T2 1 + 10.5T + 4.91e3T^{2}
19 1−58.5T+6.85e3T2 1 - 58.5T + 6.85e3T^{2}
23 1−125.T+1.21e4T2 1 - 125.T + 1.21e4T^{2}
29 1+35.4T+2.43e4T2 1 + 35.4T + 2.43e4T^{2}
31 1+291.T+2.97e4T2 1 + 291.T + 2.97e4T^{2}
37 1+259.T+5.06e4T2 1 + 259.T + 5.06e4T^{2}
41 1+338.T+6.89e4T2 1 + 338.T + 6.89e4T^{2}
43 1+6.80T+7.95e4T2 1 + 6.80T + 7.95e4T^{2}
47 1+250.T+1.03e5T2 1 + 250.T + 1.03e5T^{2}
53 1+536.T+1.48e5T2 1 + 536.T + 1.48e5T^{2}
59 1−35.8T+2.05e5T2 1 - 35.8T + 2.05e5T^{2}
61 1−57.7T+2.26e5T2 1 - 57.7T + 2.26e5T^{2}
67 1+481.T+3.00e5T2 1 + 481.T + 3.00e5T^{2}
71 1+363.T+3.57e5T2 1 + 363.T + 3.57e5T^{2}
73 1−581.T+3.89e5T2 1 - 581.T + 3.89e5T^{2}
79 1−693.T+4.93e5T2 1 - 693.T + 4.93e5T^{2}
83 1+1.33e3T+5.71e5T2 1 + 1.33e3T + 5.71e5T^{2}
89 1+353.T+7.04e5T2 1 + 353.T + 7.04e5T^{2}
97 1−1.44e3T+9.12e5T2 1 - 1.44e3T + 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

−10.55314252260507057026967221669, −9.851108601031503344667657035441, −9.035669858805245242311123142654, −7.59989857386556468797913429421, −6.73022719050793100184404417295, −5.56055115271339643577045628393, −4.98150113830503365689327347887, −3.20033352580597576395212986654, −1.82745178827283682517970098341, 0, 1.82745178827283682517970098341, 3.20033352580597576395212986654, 4.98150113830503365689327347887, 5.56055115271339643577045628393, 6.73022719050793100184404417295, 7.59989857386556468797913429421, 9.035669858805245242311123142654, 9.851108601031503344667657035441, 10.55314252260507057026967221669

Graph of the ZZ-function along the critical line