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Properties

Label 2-6552-1.1-c1-0-66
Degree 22
Conductor 65526552
Sign −1-1
Analytic cond. 52.317952.3179
Root an. cond. 7.233117.23311
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 2·11-s + 13-s − 7·19-s + 3·23-s − 4·25-s + 9·29-s + 5·31-s − 35-s − 8·37-s + 10·41-s + 5·43-s − 7·47-s + 49-s − 3·53-s − 2·55-s + 6·61-s + 65-s − 10·67-s − 4·71-s − 11·73-s + 2·77-s − 11·79-s − 11·83-s + 3·89-s − 91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.277·13-s − 1.60·19-s + 0.625·23-s − 4/5·25-s + 1.67·29-s + 0.898·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s + 0.762·43-s − 1.02·47-s + 1/7·49-s − 0.412·53-s − 0.269·55-s + 0.768·61-s + 0.124·65-s − 1.22·67-s − 0.474·71-s − 1.28·73-s + 0.227·77-s − 1.23·79-s − 1.20·83-s + 0.317·89-s − 0.104·91-s + ⋯

Functional equation

Λ(s)=(6552s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6552s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65526552    =    23⋅32⋅7⋅132^{3} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 52.317952.3179
Root analytic conductor: 7.233117.23311
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6552, ( :1/2), −1)(2,\ 6552,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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 1
7 1+T 1 + T
13 1−T 1 - T
good5 1−T+pT2 1 - T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1+11T+pT2 1 + 11 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1+15T+pT2 1 + 15 T + p 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

−7.66914847156675308748810502522, −6.83772489665389638417589680887, −6.23191861913624368915142545331, −5.66888164355744385523347126110, −4.70462681641584582462195219492, −4.12556278942285630433767071210, −3.00445400946739823266676932889, −2.41688276387460613412501930136, −1.33304308048761542490292123682, 0, 1.33304308048761542490292123682, 2.41688276387460613412501930136, 3.00445400946739823266676932889, 4.12556278942285630433767071210, 4.70462681641584582462195219492, 5.66888164355744385523347126110, 6.23191861913624368915142545331, 6.83772489665389638417589680887, 7.66914847156675308748810502522

Graph of the ZZ-function along the critical line