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Properties

Label 2-60840-1.1-c1-0-51
Degree 22
Conductor 6084060840
Sign 11
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 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·7-s − 5·11-s − 5·17-s − 19-s + 23-s + 25-s + 9·29-s − 4·31-s + 3·35-s − 3·37-s + 41-s − 3·43-s + 8·47-s + 2·49-s − 10·53-s + 5·55-s − 3·59-s + 7·61-s − 9·67-s − 7·71-s + 10·73-s + 15·77-s − 16·79-s − 12·83-s + 5·85-s − 15·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.21·17-s − 0.229·19-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.718·31-s + 0.507·35-s − 0.493·37-s + 0.156·41-s − 0.457·43-s + 1.16·47-s + 2/7·49-s − 1.37·53-s + 0.674·55-s − 0.390·59-s + 0.896·61-s − 1.09·67-s − 0.830·71-s + 1.17·73-s + 1.70·77-s − 1.80·79-s − 1.31·83-s + 0.542·85-s − 1.58·89-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: 11
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: 22
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+3T+pT2 1 + 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−T+pT2 1 - T + p T^{2}
43 1+3T+pT2 1 + 3 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+9T+pT2 1 + 9 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+15T+pT2 1 + 15 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

−14.98016483221341, −14.14193356468691, −13.78241848931467, −13.14023748375960, −12.77021653639910, −12.52294323955306, −11.76804995389109, −11.18147278613258, −10.70594833620145, −10.21966394324413, −9.832004316487263, −9.049653734191630, −8.652059512236523, −8.130249384770013, −7.459464265414493, −6.981634699499793, −6.469889939620209, −5.896755430220122, −5.224816670213413, −4.631150243027681, −4.103121638654220, −3.286744731211173, −2.793668861061841, −2.326048047574123, −1.259976587528407, 0, 0, 1.259976587528407, 2.326048047574123, 2.793668861061841, 3.286744731211173, 4.103121638654220, 4.631150243027681, 5.224816670213413, 5.896755430220122, 6.469889939620209, 6.981634699499793, 7.459464265414493, 8.130249384770013, 8.652059512236523, 9.049653734191630, 9.832004316487263, 10.21966394324413, 10.70594833620145, 11.18147278613258, 11.76804995389109, 12.52294323955306, 12.77021653639910, 13.14023748375960, 13.78241848931467, 14.14193356468691, 14.98016483221341

Graph of the ZZ-function along the critical line