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Properties

Label 2-60840-1.1-c1-0-31
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 + 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: 00
Selberg data: (2, 60840, ( :1/2), 1)(2,\ 60840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0252869604.025286960
L(12)L(\frac12) ≈\approx 4.0252869604.025286960
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

−14.36999452253120, −13.98245643511540, −13.23309197596011, −12.95866397710339, −12.09584456818377, −11.60643542024158, −11.48467512753715, −10.76138162693826, −10.25634656317411, −9.581695193615496, −9.213215957227980, −8.426213734580950, −8.371501112812523, −7.554915957360483, −6.709103629372481, −6.564702960710634, −5.976316166293907, −4.993341123084377, −4.771338464524764, −4.193295190854679, −3.454328768259540, −2.680122628644045, −1.949038155026593, −1.406354005859407, −0.7235255458280209, 0.7235255458280209, 1.406354005859407, 1.949038155026593, 2.680122628644045, 3.454328768259540, 4.193295190854679, 4.771338464524764, 4.993341123084377, 5.976316166293907, 6.564702960710634, 6.709103629372481, 7.554915957360483, 8.371501112812523, 8.426213734580950, 9.213215957227980, 9.581695193615496, 10.25634656317411, 10.76138162693826, 11.48467512753715, 11.60643542024158, 12.09584456818377, 12.95866397710339, 13.23309197596011, 13.98245643511540, 14.36999452253120

Graph of the ZZ-function along the critical line