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Properties

Label 2-336-1.1-c5-0-20
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 − 34·5-s + 49·7-s + 81·9-s + 332·11-s − 1.02e3·13-s + 306·15-s + 922·17-s − 452·19-s − 441·21-s + 3.77e3·23-s − 1.96e3·25-s − 729·27-s + 1.16e3·29-s + 9.79e3·31-s − 2.98e3·33-s − 1.66e3·35-s + 2.39e3·37-s + 9.23e3·39-s − 7.23e3·41-s − 4.65e3·43-s − 2.75e3·45-s − 2.46e4·47-s + 2.40e3·49-s − 8.29e3·51-s + 1.11e3·53-s − 1.12e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.608·5-s + 0.377·7-s + 1/3·9-s + 0.827·11-s − 1.68·13-s + 0.351·15-s + 0.773·17-s − 0.287·19-s − 0.218·21-s + 1.48·23-s − 0.630·25-s − 0.192·27-s + 0.257·29-s + 1.83·31-s − 0.477·33-s − 0.229·35-s + 0.287·37-s + 0.972·39-s − 0.671·41-s − 0.383·43-s − 0.202·45-s − 1.62·47-s + 1/7·49-s − 0.446·51-s + 0.0542·53-s − 0.503·55-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+34T+p5T2 1 + 34 T + p^{5} T^{2}
11 1−332T+p5T2 1 - 332 T + p^{5} T^{2}
13 1+1026T+p5T2 1 + 1026 T + p^{5} T^{2}
17 1−922T+p5T2 1 - 922 T + p^{5} T^{2}
19 1+452T+p5T2 1 + 452 T + p^{5} T^{2}
23 1−3776T+p5T2 1 - 3776 T + p^{5} T^{2}
29 1−1166T+p5T2 1 - 1166 T + p^{5} T^{2}
31 1−9792T+p5T2 1 - 9792 T + p^{5} T^{2}
37 1−2390T+p5T2 1 - 2390 T + p^{5} T^{2}
41 1+7230T+p5T2 1 + 7230 T + p^{5} T^{2}
43 1+4652T+p5T2 1 + 4652 T + p^{5} T^{2}
47 1+24672T+p5T2 1 + 24672 T + p^{5} T^{2}
53 1−1110T+p5T2 1 - 1110 T + p^{5} T^{2}
59 1+46892T+p5T2 1 + 46892 T + p^{5} T^{2}
61 1+9762T+p5T2 1 + 9762 T + p^{5} T^{2}
67 1−26252T+p5T2 1 - 26252 T + p^{5} T^{2}
71 1+65440T+p5T2 1 + 65440 T + p^{5} T^{2}
73 1+5606T+p5T2 1 + 5606 T + p^{5} T^{2}
79 1−9840T+p5T2 1 - 9840 T + p^{5} T^{2}
83 1+61108T+p5T2 1 + 61108 T + p^{5} T^{2}
89 1+62958T+p5T2 1 + 62958 T + p^{5} T^{2}
97 1+37838T+p5T2 1 + 37838 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

−10.25856883851818674074313788294, −9.487309125731153278184453242928, −8.231654937444945741664297685900, −7.34548424296612183582124217320, −6.45795571312658408086281263883, −5.11008873504651754148495748494, −4.38399259143924650933903289331, −2.95600532809480101128262969901, −1.32791277727057133022615217788, 0, 1.32791277727057133022615217788, 2.95600532809480101128262969901, 4.38399259143924650933903289331, 5.11008873504651754148495748494, 6.45795571312658408086281263883, 7.34548424296612183582124217320, 8.231654937444945741664297685900, 9.487309125731153278184453242928, 10.25856883851818674074313788294

Graph of the ZZ-function along the critical line