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Properties

Label 2-336-1.1-c3-0-12
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 − 4.54·5-s − 7·7-s + 9·9-s + 40.7·11-s + 53.2·13-s + 13.6·15-s + 4.54·17-s − 122.·19-s + 21·21-s − 131.·23-s − 104.·25-s − 27·27-s − 216.·29-s + 251.·31-s − 122.·33-s + 31.8·35-s + 11.8·37-s − 159.·39-s − 111.·41-s − 369.·43-s − 40.9·45-s + 262.·47-s + 49·49-s − 13.6·51-s − 567.·53-s − 185.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.406·5-s − 0.377·7-s + 0.333·9-s + 1.11·11-s + 1.13·13-s + 0.234·15-s + 0.0649·17-s − 1.48·19-s + 0.218·21-s − 1.19·23-s − 0.834·25-s − 0.192·27-s − 1.38·29-s + 1.45·31-s − 0.644·33-s + 0.153·35-s + 0.0528·37-s − 0.656·39-s − 0.425·41-s − 1.30·43-s − 0.135·45-s + 0.815·47-s + 0.142·49-s − 0.0374·51-s − 1.46·53-s − 0.454·55-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+4.54T+125T2 1 + 4.54T + 125T^{2}
11 1−40.7T+1.33e3T2 1 - 40.7T + 1.33e3T^{2}
13 1−53.2T+2.19e3T2 1 - 53.2T + 2.19e3T^{2}
17 1−4.54T+4.91e3T2 1 - 4.54T + 4.91e3T^{2}
19 1+122.T+6.85e3T2 1 + 122.T + 6.85e3T^{2}
23 1+131.T+1.21e4T2 1 + 131.T + 1.21e4T^{2}
29 1+216.T+2.43e4T2 1 + 216.T + 2.43e4T^{2}
31 1−251.T+2.97e4T2 1 - 251.T + 2.97e4T^{2}
37 1−11.8T+5.06e4T2 1 - 11.8T + 5.06e4T^{2}
41 1+111.T+6.89e4T2 1 + 111.T + 6.89e4T^{2}
43 1+369.T+7.95e4T2 1 + 369.T + 7.95e4T^{2}
47 1−262.T+1.03e5T2 1 - 262.T + 1.03e5T^{2}
53 1+567.T+1.48e5T2 1 + 567.T + 1.48e5T^{2}
59 1+839.T+2.05e5T2 1 + 839.T + 2.05e5T^{2}
61 1+485.T+2.26e5T2 1 + 485.T + 2.26e5T^{2}
67 1−333.T+3.00e5T2 1 - 333.T + 3.00e5T^{2}
71 1+590.T+3.57e5T2 1 + 590.T + 3.57e5T^{2}
73 1−490.T+3.89e5T2 1 - 490.T + 3.89e5T^{2}
79 1+121.T+4.93e5T2 1 + 121.T + 4.93e5T^{2}
83 1+609.T+5.71e5T2 1 + 609.T + 5.71e5T^{2}
89 1−719.T+7.04e5T2 1 - 719.T + 7.04e5T^{2}
97 1+637.T+9.12e5T2 1 + 637.T + 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

−10.79362011636077056360559561261, −9.808679348015717556555181655322, −8.785405480726387599516454570238, −7.82202061269941747277692550723, −6.46557243205676936061600349675, −6.04793192761779134497314478525, −4.40433686169121043609948429682, −3.61100062040112321213645830065, −1.65353758962625653150982339879, 0, 1.65353758962625653150982339879, 3.61100062040112321213645830065, 4.40433686169121043609948429682, 6.04793192761779134497314478525, 6.46557243205676936061600349675, 7.82202061269941747277692550723, 8.785405480726387599516454570238, 9.808679348015717556555181655322, 10.79362011636077056360559561261

Graph of the ZZ-function along the critical line