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Properties

Label 2-336-1.1-c5-0-29
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 + 78·5-s − 49·7-s + 81·9-s − 444·11-s − 442·13-s + 702·15-s − 126·17-s − 2.68e3·19-s − 441·21-s − 4.20e3·23-s + 2.95e3·25-s + 729·27-s − 5.44e3·29-s − 80·31-s − 3.99e3·33-s − 3.82e3·35-s − 5.43e3·37-s − 3.97e3·39-s + 7.96e3·41-s + 1.15e4·43-s + 6.31e3·45-s + 1.39e4·47-s + 2.40e3·49-s − 1.13e3·51-s − 9.59e3·53-s − 3.46e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.39·5-s − 0.377·7-s + 1/3·9-s − 1.10·11-s − 0.725·13-s + 0.805·15-s − 0.105·17-s − 1.70·19-s − 0.218·21-s − 1.65·23-s + 0.946·25-s + 0.192·27-s − 1.20·29-s − 0.0149·31-s − 0.638·33-s − 0.527·35-s − 0.652·37-s − 0.418·39-s + 0.739·41-s + 0.950·43-s + 0.465·45-s + 0.919·47-s + 1/7·49-s − 0.0610·51-s − 0.469·53-s − 1.54·55-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 1−78T+p5T2 1 - 78 T + p^{5} T^{2}
11 1+444T+p5T2 1 + 444 T + p^{5} T^{2}
13 1+34pT+p5T2 1 + 34 p T + p^{5} T^{2}
17 1+126T+p5T2 1 + 126 T + p^{5} T^{2}
19 1+2684T+p5T2 1 + 2684 T + p^{5} T^{2}
23 1+4200T+p5T2 1 + 4200 T + p^{5} T^{2}
29 1+5442T+p5T2 1 + 5442 T + p^{5} T^{2}
31 1+80T+p5T2 1 + 80 T + p^{5} T^{2}
37 1+5434T+p5T2 1 + 5434 T + p^{5} T^{2}
41 1−7962T+p5T2 1 - 7962 T + p^{5} T^{2}
43 1−268pT+p5T2 1 - 268 p T + p^{5} T^{2}
47 1−13920T+p5T2 1 - 13920 T + p^{5} T^{2}
53 1+9594T+p5T2 1 + 9594 T + p^{5} T^{2}
59 1+27492T+p5T2 1 + 27492 T + p^{5} T^{2}
61 1−49478T+p5T2 1 - 49478 T + p^{5} T^{2}
67 1−59356T+p5T2 1 - 59356 T + p^{5} T^{2}
71 1+32040T+p5T2 1 + 32040 T + p^{5} T^{2}
73 1+61846T+p5T2 1 + 61846 T + p^{5} T^{2}
79 1−65776T+p5T2 1 - 65776 T + p^{5} T^{2}
83 1+40188T+p5T2 1 + 40188 T + p^{5} T^{2}
89 1+7974T+p5T2 1 + 7974 T + p^{5} T^{2}
97 1+143662T+p5T2 1 + 143662 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

−10.10430115718847556130352365208, −9.475976354722215030326769534513, −8.471816156807295115973758140692, −7.43480032203906778389900176978, −6.25192840392640455084044631646, −5.44475809352900994440653809888, −4.11105977679246028849055617928, −2.52697664561500972094960474988, −1.98735456118617454596188627028, 0, 1.98735456118617454596188627028, 2.52697664561500972094960474988, 4.11105977679246028849055617928, 5.44475809352900994440653809888, 6.25192840392640455084044631646, 7.43480032203906778389900176978, 8.471816156807295115973758140692, 9.475976354722215030326769534513, 10.10430115718847556130352365208

Graph of the ZZ-function along the critical line