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Properties

Label 2-60840-1.1-c1-0-4
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 4·11-s + 8·17-s + 2·19-s + 4·23-s + 25-s − 8·29-s − 10·31-s + 2·35-s − 6·37-s − 6·41-s − 8·43-s + 8·47-s − 3·49-s − 12·53-s + 4·55-s + 4·59-s + 10·61-s − 2·67-s − 6·73-s + 8·77-s + 12·79-s + 4·83-s − 8·85-s + 6·89-s − 2·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 1.20·11-s + 1.94·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.79·31-s + 0.338·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.244·67-s − 0.702·73-s + 0.911·77-s + 1.35·79-s + 0.439·83-s − 0.867·85-s + 0.635·89-s − 0.205·95-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: 00
Selberg data: (2, 60840, ( :1/2), 1)(2,\ 60840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.78031375230.7803137523
L(12)L(\frac12) ≈\approx 0.78031375230.7803137523
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+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−12T+pT2 1 - 12 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

−14.38083691831185, −13.74598796079006, −13.08531584788219, −12.88506750881231, −12.34959733066861, −11.82242275428402, −11.23698840688485, −10.69327177858312, −10.22491091825306, −9.710995064937718, −9.241403229592174, −8.616207558172191, −7.923702419564040, −7.542310860384827, −7.138871381098519, −6.468097766995289, −5.606251857101731, −5.336416722979387, −4.891550262475783, −3.682916351930839, −3.493593527000839, −2.998721373616594, −2.081280250045705, −1.310188927303040, −0.3070367252841509, 0.3070367252841509, 1.310188927303040, 2.081280250045705, 2.998721373616594, 3.493593527000839, 3.682916351930839, 4.891550262475783, 5.336416722979387, 5.606251857101731, 6.468097766995289, 7.138871381098519, 7.542310860384827, 7.923702419564040, 8.616207558172191, 9.241403229592174, 9.710995064937718, 10.22491091825306, 10.69327177858312, 11.23698840688485, 11.82242275428402, 12.34959733066861, 12.88506750881231, 13.08531584788219, 13.74598796079006, 14.38083691831185

Graph of the ZZ-function along the critical line