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Properties

Label 2-540-1.1-c1-0-2
Degree 22
Conductor 540540
Sign 11
Analytic cond. 4.311924.31192
Root an. cond. 2.076512.07651
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 2·13-s − 3·17-s + 5·19-s + 3·23-s + 25-s − 6·29-s + 5·31-s + 2·35-s + 2·37-s + 12·41-s + 8·43-s − 12·47-s − 3·49-s − 3·53-s + 6·59-s − 7·61-s + 2·65-s + 2·67-s + 12·71-s − 16·73-s − 79-s − 15·83-s − 3·85-s − 12·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.554·13-s − 0.727·17-s + 1.14·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s + 0.898·31-s + 0.338·35-s + 0.328·37-s + 1.87·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.412·53-s + 0.781·59-s − 0.896·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s − 1.87·73-s − 0.112·79-s − 1.64·83-s − 0.325·85-s − 1.27·89-s + 0.419·91-s + ⋯

Functional equation

Λ(s)=(540s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(540s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 540540    =    22⋅33⋅52^{2} \cdot 3^{3} \cdot 5
Sign: 11
Analytic conductor: 4.311924.31192
Root analytic conductor: 2.076512.07651
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 540, ( :1/2), 1)(2,\ 540,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7249938231.724993823
L(12)L(\frac12) ≈\approx 1.7249938231.724993823
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
5 1−T 1 - T
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−12T+pT2 1 - 12 T + 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+3T+pT2 1 + 3 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+16T+pT2 1 + 16 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−11.10209444324982789310858231565, −9.842081869011709127474857766049, −9.115792097624162292443271966817, −8.154384332237623931237910816096, −7.26654410829263861371133387511, −6.15894405862047630422981455568, −5.23560056211977793668529426396, −4.20739540356639782399975882305, −2.78350811936244842243504830094, −1.37424603958686381672097450800, 1.37424603958686381672097450800, 2.78350811936244842243504830094, 4.20739540356639782399975882305, 5.23560056211977793668529426396, 6.15894405862047630422981455568, 7.26654410829263861371133387511, 8.154384332237623931237910816096, 9.115792097624162292443271966817, 9.842081869011709127474857766049, 11.10209444324982789310858231565

Graph of the ZZ-function along the critical line