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Properties

Label 2-3240-1.1-c1-0-38
Degree 22
Conductor 32403240
Sign −1-1
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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 − 2·7-s − 3·11-s + 2·17-s + 19-s + 2·23-s + 25-s − 3·29-s − 3·31-s − 2·35-s + 5·41-s − 4·43-s − 8·47-s − 3·49-s + 2·53-s − 3·55-s + 3·59-s + 6·61-s − 10·67-s − 15·71-s − 14·73-s + 6·77-s − 8·79-s + 2·85-s + 89-s + 95-s − 16·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.904·11-s + 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s − 0.338·35-s + 0.780·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.404·55-s + 0.390·59-s + 0.768·61-s − 1.22·67-s − 1.78·71-s − 1.63·73-s + 0.683·77-s − 0.900·79-s + 0.216·85-s + 0.105·89-s + 0.102·95-s − 1.62·97-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: −1-1
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3240, ( :1/2), −1)(2,\ 3240,\ (\ :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
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+15T+pT2 1 + 15 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−T+pT2 1 - T + p T^{2}
97 1+16T+pT2 1 + 16 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

−8.296182208447983126213987155467, −7.46931054670746793463002543690, −6.83407958246025426756264582364, −5.88753911467151255632721223678, −5.40064086492557480332722582663, −4.43154984924062291000787946753, −3.32721269409746942774135228772, −2.69549738228437653861646620092, −1.50342828949796868296464122084, 0, 1.50342828949796868296464122084, 2.69549738228437653861646620092, 3.32721269409746942774135228772, 4.43154984924062291000787946753, 5.40064086492557480332722582663, 5.88753911467151255632721223678, 6.83407958246025426756264582364, 7.46931054670746793463002543690, 8.296182208447983126213987155467

Graph of the ZZ-function along the critical line