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Properties

Label 4-1216e2-1.1-c1e2-0-5
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  + 2·5-s − 6·9-s + 14·17-s − 7·25-s − 12·45-s − 5·49-s − 30·61-s + 22·73-s + 27·81-s + 28·85-s + 20·101-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 84·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 3.39·17-s − 7/5·25-s − 1.78·45-s − 5/7·49-s − 3.84·61-s + 2.57·73-s + 3·81-s + 3.03·85-s + 1.99·101-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 6.79·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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 2.0143515252.014351525
L(12)L(\frac12) ≈\approx 2.0143515252.014351525
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+pT2 1 + p T^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
47C2C_2 (1−13T+pT2)(1+13T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} )
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
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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

−9.733159759828253484517382507585, −9.598640470651873021927810079484, −9.289689289756409509468176085632, −8.730652112905267376001911698192, −8.182529881711780159313974762446, −7.892461696995965956192970964650, −7.72824453972356422499019613261, −7.14724539298870820808912222030, −6.27600795084745381564055865545, −6.05933519784711203569568899208, −5.84570053584007553397729896467, −5.27621839607647452944859918194, −5.19137696551245104377824290314, −4.36506464712612395828591044825, −3.44894845950579846685311982810, −3.33993944117661109114552992065, −2.88082250231297329570457088281, −2.10492350651637610444647579425, −1.54550165955808104194140134970, −0.61090724662325347051808232074, 0.61090724662325347051808232074, 1.54550165955808104194140134970, 2.10492350651637610444647579425, 2.88082250231297329570457088281, 3.33993944117661109114552992065, 3.44894845950579846685311982810, 4.36506464712612395828591044825, 5.19137696551245104377824290314, 5.27621839607647452944859918194, 5.84570053584007553397729896467, 6.05933519784711203569568899208, 6.27600795084745381564055865545, 7.14724539298870820808912222030, 7.72824453972356422499019613261, 7.892461696995965956192970964650, 8.182529881711780159313974762446, 8.730652112905267376001911698192, 9.289689289756409509468176085632, 9.598640470651873021927810079484, 9.733159759828253484517382507585

Graph of the ZZ-function along the critical line