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Properties

Label 2-5040-1.1-c1-0-15
Degree 22
Conductor 50405040
Sign 11
Analytic cond. 40.244640.2446
Root an. cond. 6.343866.34386
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 + 7-s − 5·11-s + 13-s − 3·17-s + 6·19-s − 6·23-s + 25-s + 9·29-s + 35-s + 6·37-s − 8·41-s − 6·43-s + 3·47-s + 49-s + 12·53-s − 5·55-s + 8·59-s − 4·61-s + 65-s + 4·67-s + 8·71-s + 10·73-s − 5·77-s + 3·79-s − 12·83-s − 3·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.277·13-s − 0.727·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.169·35-s + 0.986·37-s − 1.24·41-s − 0.914·43-s + 0.437·47-s + 1/7·49-s + 1.64·53-s − 0.674·55-s + 1.04·59-s − 0.512·61-s + 0.124·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.569·77-s + 0.337·79-s − 1.31·83-s − 0.325·85-s + ⋯

Functional equation

Λ(s)=(5040s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5040s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 50405040    =    24⋅32⋅5⋅72^{4} \cdot 3^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 40.244640.2446
Root analytic conductor: 6.343866.34386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5040, ( :1/2), 1)(2,\ 5040,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9796811151.979681115
L(12)L(\frac12) ≈\approx 1.9796811151.979681115
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
7 1−T 1 - T
good11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−3T+pT2 1 - 3 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−16T+pT2 1 - 16 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

−8.221402281878789465089589313507, −7.62954374641588235718006492065, −6.79230502198732864314633745798, −6.01444826736210837468924519845, −5.24154681791164573770536092994, −4.76706705720314731476536835342, −3.68523744465265841556690270503, −2.71239320609283552284776322612, −2.03744261910962500613971262945, −0.76044480432284251382048398807, 0.76044480432284251382048398807, 2.03744261910962500613971262945, 2.71239320609283552284776322612, 3.68523744465265841556690270503, 4.76706705720314731476536835342, 5.24154681791164573770536092994, 6.01444826736210837468924519845, 6.79230502198732864314633745798, 7.62954374641588235718006492065, 8.221402281878789465089589313507

Graph of the ZZ-function along the critical line