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Properties

Label 2-1680-1.1-c3-0-66
Degree 22
Conductor 16801680
Sign −1-1
Analytic cond. 99.123299.1232
Root an. cond. 9.956069.95606
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·5-s − 7·7-s + 9·9-s − 12·11-s + 30·13-s + 15·15-s − 134·17-s + 92·19-s − 21·21-s − 112·23-s + 25·25-s + 27·27-s − 58·29-s + 224·31-s − 36·33-s − 35·35-s − 146·37-s + 90·39-s + 18·41-s − 340·43-s + 45·45-s − 208·47-s + 49·49-s − 402·51-s − 754·53-s − 60·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.328·11-s + 0.640·13-s + 0.258·15-s − 1.91·17-s + 1.11·19-s − 0.218·21-s − 1.01·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.29·31-s − 0.189·33-s − 0.169·35-s − 0.648·37-s + 0.369·39-s + 0.0685·41-s − 1.20·43-s + 0.149·45-s − 0.645·47-s + 1/7·49-s − 1.10·51-s − 1.95·53-s − 0.147·55-s + ⋯

Functional equation

Λ(s)=(1680s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1680s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16801680    =    24⋅3⋅5⋅72^{4} \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 99.123299.1232
Root analytic conductor: 9.956069.95606
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1680, ( :3/2), −1)(2,\ 1680,\ (\ :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−pT 1 - p T
5 1−pT 1 - p T
7 1+pT 1 + p T
good11 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
17 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
19 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
23 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
29 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
31 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
37 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
41 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
43 1+340T+p3T2 1 + 340 T + p^{3} T^{2}
47 1+208T+p3T2 1 + 208 T + p^{3} T^{2}
53 1+754T+p3T2 1 + 754 T + p^{3} T^{2}
59 1+380T+p3T2 1 + 380 T + p^{3} T^{2}
61 1−718T+p3T2 1 - 718 T + p^{3} T^{2}
67 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
71 1−960T+p3T2 1 - 960 T + p^{3} T^{2}
73 1−1066T+p3T2 1 - 1066 T + p^{3} T^{2}
79 1+896T+p3T2 1 + 896 T + p^{3} T^{2}
83 1+436T+p3T2 1 + 436 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1+702T+p3T2 1 + 702 T + p^{3} 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

−8.543481349741917636047345990590, −8.010990647727093230688422972333, −6.85331684118774639790377385827, −6.35881672344508460526435650590, −5.29009273179942624697250231256, −4.35103005648574436351837721012, −3.37514022408927255250730284216, −2.46379457244669572329142236479, −1.49755028586111857837268868380, 0, 1.49755028586111857837268868380, 2.46379457244669572329142236479, 3.37514022408927255250730284216, 4.35103005648574436351837721012, 5.29009273179942624697250231256, 6.35881672344508460526435650590, 6.85331684118774639790377385827, 8.010990647727093230688422972333, 8.543481349741917636047345990590

Graph of the ZZ-function along the critical line