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Properties

Label 4-1216e2-1.1-c1e2-0-11
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 3·9-s + 9·13-s − 3·15-s − 3·17-s − 8·19-s + 9·23-s + 5·25-s − 8·27-s − 15·29-s − 8·31-s − 9·39-s + 15·41-s + 21·43-s + 9·45-s − 3·47-s + 2·49-s + 3·51-s − 3·53-s + 8·57-s + 3·59-s + 7·61-s + 27·65-s + 5·67-s − 9·69-s + 9·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 9-s + 2.49·13-s − 0.774·15-s − 0.727·17-s − 1.83·19-s + 1.87·23-s + 25-s − 1.53·27-s − 2.78·29-s − 1.43·31-s − 1.44·39-s + 2.34·41-s + 3.20·43-s + 1.34·45-s − 0.437·47-s + 2/7·49-s + 0.420·51-s − 0.412·53-s + 1.05·57-s + 0.390·59-s + 0.896·61-s + 3.34·65-s + 0.610·67-s − 1.08·69-s + 1.06·71-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7804645012.780464501
L(12)L(\frac12) ≈\approx 2.7804645012.780464501
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
19C2C_2 1+8T+pT2 1 + 8 T + p T^{2}
good3C22C_2^2 1+T−2T2+pT3+p2T4 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}
5C22C_2^2 1−3T+4T2−3pT3+p2T4 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−7T+pT2)(1−2T+pT2) ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} )
17C22C_2^2 1+3T−8T2+3pT3+p2T4 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−9T+50T2−9pT3+p2T4 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+15T+104T2+15pT3+p2T4 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
41C22C_2^2 1−15T+116T2−15pT3+p2T4 1 - 15 T + 116 T^{2} - 15 p T^{3} + p^{2} T^{4}
43C2C_2 (1−13T+pT2)(1−8T+pT2) ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} )
47C22C_2^2 1+3T+50T2+3pT3+p2T4 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+3T+56T2+3pT3+p2T4 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−3T−50T2−3pT3+p2T4 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−7T−12T2−7pT3+p2T4 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4}
67C2C_2 (1−16T+pT2)(1+11T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} )
71C22C_2^2 1−9T+10T2−9pT3+p2T4 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)(1+17T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C22C_2^2 1+7T−30T2+7pT3+p2T4 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−154T2+p2T4 1 - 154 T^{2} + p^{2} T^{4}
89C22C_2^2 1−15T+164T2−15pT3+p2T4 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−15T+172T2−15pT3+p2T4 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.724165523659296014536123864329, −9.567417471854815428389699516948, −9.026401424049172458813536877265, −8.906913540367312830580931253511, −8.625195403831133538920950547536, −7.57952143238493308166712039607, −7.56135636124548621508590370800, −7.00858219441326338529177599117, −6.32311778038392638116919414732, −6.22964477203102781360121394165, −5.75115739975073853379409614386, −5.55352417425657020935876995112, −4.91461467707331657764700480005, −4.07566690105295550910967944583, −4.03490186099514628124440792627, −3.51936560331470584323808025152, −2.44365474790947137102396523328, −2.06138151206960961211917415235, −1.49797549079084728521024767004, −0.78678816353653160920299950479, 0.78678816353653160920299950479, 1.49797549079084728521024767004, 2.06138151206960961211917415235, 2.44365474790947137102396523328, 3.51936560331470584323808025152, 4.03490186099514628124440792627, 4.07566690105295550910967944583, 4.91461467707331657764700480005, 5.55352417425657020935876995112, 5.75115739975073853379409614386, 6.22964477203102781360121394165, 6.32311778038392638116919414732, 7.00858219441326338529177599117, 7.56135636124548621508590370800, 7.57952143238493308166712039607, 8.625195403831133538920950547536, 8.906913540367312830580931253511, 9.026401424049172458813536877265, 9.567417471854815428389699516948, 9.724165523659296014536123864329

Graph of the ZZ-function along the critical line