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Properties

Label 2-4140-1.1-c1-0-6
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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 + 4·11-s + 13-s − 4·19-s + 23-s + 25-s + 7·29-s − 7·31-s + 2·35-s − 4·37-s − 3·41-s + 6·43-s + 13·47-s − 3·49-s − 10·53-s − 4·55-s + 8·59-s − 65-s + 8·67-s − 13·71-s + 11·73-s − 8·77-s + 4·79-s + 4·83-s + 6·89-s − 2·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.29·29-s − 1.25·31-s + 0.338·35-s − 0.657·37-s − 0.468·41-s + 0.914·43-s + 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s + 1.04·59-s − 0.124·65-s + 0.977·67-s − 1.54·71-s + 1.28·73-s − 0.911·77-s + 0.450·79-s + 0.439·83-s + 0.635·89-s − 0.209·91-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5307411491.530741149
L(12)L(\frac12) ≈\approx 1.5307411491.530741149
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
23 1−T 1 - T
good7 1+2T+pT2 1 + 2 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+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 1−7T+pT2 1 - 7 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+13T+pT2 1 + 13 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.582977332348146562710091043065, −7.64570064973687093757758439724, −6.79272605407006166458827493072, −6.42501841167079437554065819928, −5.53392590775861732995702419988, −4.46792999426338875485798833452, −3.82744709866793850051848950489, −3.10614689114143817615345596705, −1.94178657204270397883871050919, −0.70427278926568134058720289386, 0.70427278926568134058720289386, 1.94178657204270397883871050919, 3.10614689114143817615345596705, 3.82744709866793850051848950489, 4.46792999426338875485798833452, 5.53392590775861732995702419988, 6.42501841167079437554065819928, 6.79272605407006166458827493072, 7.64570064973687093757758439724, 8.582977332348146562710091043065

Graph of the ZZ-function along the critical line