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Properties

Label 2-6e3-1.1-c1-0-3
Degree 22
Conductor 216216
Sign −1-1
Analytic cond. 1.724761.72476
Root an. cond. 1.313301.31330
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  − 4·5-s − 3·7-s − 4·11-s + 13-s + 4·17-s − 19-s − 4·23-s + 11·25-s − 4·31-s + 12·35-s − 9·37-s − 8·43-s + 12·47-s + 2·49-s + 8·53-s + 16·55-s − 4·59-s − 5·61-s − 4·65-s + 11·67-s − 8·71-s + 73-s + 12·77-s − 5·79-s − 8·83-s − 16·85-s − 12·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.13·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 11/5·25-s − 0.718·31-s + 2.02·35-s − 1.47·37-s − 1.21·43-s + 1.75·47-s + 2/7·49-s + 1.09·53-s + 2.15·55-s − 0.520·59-s − 0.640·61-s − 0.496·65-s + 1.34·67-s − 0.949·71-s + 0.117·73-s + 1.36·77-s − 0.562·79-s − 0.878·83-s − 1.73·85-s − 1.27·89-s + ⋯

Functional equation

Λ(s)=(216s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(216s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 216216    =    23⋅332^{3} \cdot 3^{3}
Sign: −1-1
Analytic conductor: 1.724761.72476
Root analytic conductor: 1.313301.31330
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 216, ( :1/2), −1)(2,\ 216,\ (\ :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
good5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+9T+pT2 1 + 9 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−8T+pT2 1 - 8 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1+5T+pT2 1 + 5 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−5T+pT2 1 - 5 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

−12.03544564786177544377995408667, −10.86882134051916494593969688660, −10.05433643313669533911783232246, −8.655801172150449552881383118942, −7.79497553082532979201942816895, −6.95550481778817521461633350929, −5.48459428248370426498378218555, −3.99250331289836498435902686175, −3.09788365205681643264055516753, 0, 3.09788365205681643264055516753, 3.99250331289836498435902686175, 5.48459428248370426498378218555, 6.95550481778817521461633350929, 7.79497553082532979201942816895, 8.655801172150449552881383118942, 10.05433643313669533911783232246, 10.86882134051916494593969688660, 12.03544564786177544377995408667

Graph of the ZZ-function along the critical line