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Properties

Label 2-60840-1.1-c1-0-38
Degree 22
Conductor 6084060840
Sign −1-1
Analytic cond. 485.809485.809
Root an. cond. 22.041022.0410
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 − 6·17-s − 8·19-s + 2·23-s + 25-s + 8·29-s − 7·31-s + 35-s + 2·37-s + 6·41-s − 43-s + 8·47-s − 6·49-s − 4·53-s − 2·55-s − 8·59-s + 7·61-s + 67-s + 8·71-s + 13·73-s − 2·77-s + 5·79-s + 12·83-s − 6·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.45·17-s − 1.83·19-s + 0.417·23-s + 1/5·25-s + 1.48·29-s − 1.25·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s − 0.549·53-s − 0.269·55-s − 1.04·59-s + 0.896·61-s + 0.122·67-s + 0.949·71-s + 1.52·73-s − 0.227·77-s + 0.562·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s + ⋯

Functional equation

Λ(s)=(60840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(60840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6084060840    =    23⋅32⋅5⋅1322^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 485.809485.809
Root analytic conductor: 22.041022.0410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 60840, ( :1/2), −1)(2,\ 60840,\ (\ :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
5 1−T 1 - T
13 1 1
good7 1−T+pT2 1 - T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−13T+pT2 1 - 13 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−14.45703132346845, −14.10979610537037, −13.46004531588798, −13.06249351713960, −12.56338894282909, −12.23520905619911, −11.24434544941690, −10.86487301847849, −10.75691169563360, −9.976340090320831, −9.337900624113648, −8.910703535130298, −8.336830016763164, −7.965360985758222, −7.167166915693500, −6.579121028942860, −6.275579212673327, −5.526171333716326, −4.872562758344399, −4.470223785086471, −3.863290703704856, −2.953114631327286, −2.234557200117842, −2.000708022356538, −0.8990816307910405, 0, 0.8990816307910405, 2.000708022356538, 2.234557200117842, 2.953114631327286, 3.863290703704856, 4.470223785086471, 4.872562758344399, 5.526171333716326, 6.275579212673327, 6.579121028942860, 7.167166915693500, 7.965360985758222, 8.336830016763164, 8.910703535130298, 9.337900624113648, 9.976340090320831, 10.75691169563360, 10.86487301847849, 11.24434544941690, 12.23520905619911, 12.56338894282909, 13.06249351713960, 13.46004531588798, 14.10979610537037, 14.45703132346845

Graph of the ZZ-function along the critical line