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Properties

Label 2-84-1.1-c5-0-4
Degree 22
Conductor 8484
Sign −1-1
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
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 + 6·5-s + 49·7-s + 81·9-s − 108·11-s − 346·13-s − 54·15-s − 1.39e3·17-s − 1.01e3·19-s − 441·21-s − 1.53e3·23-s − 3.08e3·25-s − 729·27-s − 3.76e3·29-s − 736·31-s + 972·33-s + 294·35-s + 2.05e3·37-s + 3.11e3·39-s − 1.55e4·41-s + 1.10e4·43-s + 486·45-s + 4.56e3·47-s + 2.40e3·49-s + 1.25e4·51-s − 7.96e3·53-s − 648·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.107·5-s + 0.377·7-s + 1/3·9-s − 0.269·11-s − 0.567·13-s − 0.0619·15-s − 1.17·17-s − 0.643·19-s − 0.218·21-s − 0.605·23-s − 0.988·25-s − 0.192·27-s − 0.830·29-s − 0.137·31-s + 0.155·33-s + 0.0405·35-s + 0.246·37-s + 0.327·39-s − 1.44·41-s + 0.910·43-s + 0.0357·45-s + 0.301·47-s + 1/7·49-s + 0.677·51-s − 0.389·53-s − 0.0288·55-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 84, ( :5/2), −1)(2,\ 84,\ (\ :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 1−6T+p5T2 1 - 6 T + p^{5} T^{2}
11 1+108T+p5T2 1 + 108 T + p^{5} T^{2}
13 1+346T+p5T2 1 + 346 T + p^{5} T^{2}
17 1+1398T+p5T2 1 + 1398 T + p^{5} T^{2}
19 1+1012T+p5T2 1 + 1012 T + p^{5} T^{2}
23 1+1536T+p5T2 1 + 1536 T + p^{5} T^{2}
29 1+3762T+p5T2 1 + 3762 T + p^{5} T^{2}
31 1+736T+p5T2 1 + 736 T + p^{5} T^{2}
37 1−2054T+p5T2 1 - 2054 T + p^{5} T^{2}
41 1+15534T+p5T2 1 + 15534 T + p^{5} T^{2}
43 1−11036T+p5T2 1 - 11036 T + p^{5} T^{2}
47 1−4560T+p5T2 1 - 4560 T + p^{5} T^{2}
53 1+7962T+p5T2 1 + 7962 T + p^{5} T^{2}
59 1+7020T+p5T2 1 + 7020 T + p^{5} T^{2}
61 1−26870T+p5T2 1 - 26870 T + p^{5} T^{2}
67 1−52148T+p5T2 1 - 52148 T + p^{5} T^{2}
71 1+2544T+p5T2 1 + 2544 T + p^{5} T^{2}
73 1+9766T+p5T2 1 + 9766 T + p^{5} T^{2}
79 1−68672T+p5T2 1 - 68672 T + p^{5} T^{2}
83 1+61668T+p5T2 1 + 61668 T + p^{5} T^{2}
89 1+41454T+p5T2 1 + 41454 T + p^{5} T^{2}
97 1+111262T+p5T2 1 + 111262 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

−12.70987724973401599935095432225, −11.61040791791882310951281541863, −10.69033219718494232263830482518, −9.521831963978602286033636957168, −8.124799006866335743308090813313, −6.82080511089064574446156943468, −5.51957437080052593999409368883, −4.22206514312905283403621173119, −2.07201297117303168597525594645, 0, 2.07201297117303168597525594645, 4.22206514312905283403621173119, 5.51957437080052593999409368883, 6.82080511089064574446156943468, 8.124799006866335743308090813313, 9.521831963978602286033636957168, 10.69033219718494232263830482518, 11.61040791791882310951281541863, 12.70987724973401599935095432225

Graph of the ZZ-function along the critical line