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Properties

Label 4-1027e2-1.1-c0e2-0-0
Degree 44
Conductor 10547291054729
Sign 11
Analytic cond. 0.2626970.262697
Root an. cond. 0.7159180.715918
Motivic weight 00
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-s + 4-s − 2·5-s + 2·8-s − 9-s − 2·10-s + 11-s + 2·13-s + 2·16-s − 18-s + 19-s − 2·20-s + 22-s + 23-s + 25-s + 2·26-s − 2·31-s + 2·32-s − 36-s + 38-s − 4·40-s + 44-s + 2·45-s + 46-s − 49-s + 50-s + 2·52-s + ⋯
L(s)  = 1  + 2-s + 4-s − 2·5-s + 2·8-s − 9-s − 2·10-s + 11-s + 2·13-s + 2·16-s − 18-s + 19-s − 2·20-s + 22-s + 23-s + 25-s + 2·26-s − 2·31-s + 2·32-s − 36-s + 38-s − 4·40-s + 44-s + 2·45-s + 46-s − 49-s + 50-s + 2·52-s + ⋯

Functional equation

Λ(s)=(1054729s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1054729s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10547291054729    =    132⋅79213^{2} \cdot 79^{2}
Sign: 11
Analytic conductor: 0.2626970.262697
Root analytic conductor: 0.7159180.715918
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1054729, ( :0,0), 1)(4,\ 1054729,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6231838181.623183818
L(12)L(\frac12) ≈\approx 1.6231838181.623183818
L(1)L(1) 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)
bad13C1C_1 (1−T)2 ( 1 - T )^{2}
79C1C_1 (1−T)2 ( 1 - T )^{2}
good2C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
3C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
7C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
11C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
23C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
37C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
71C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
73C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
83C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
89C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
97C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{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

−10.46615755009783933935676078390, −10.27110015600269272082278324988, −9.335569883374458015250617223554, −9.063305693524270150017827572523, −8.391963279391228760465524712718, −8.380450659643299525304038593209, −7.56844212499678880318083819740, −7.53059444797042939960776823323, −7.10612424694351373910820587978, −6.52686038307702376327906944098, −6.03885496050121979241949048776, −5.59579701489961261337259153152, −5.15060879542347511279056135005, −4.39647740261656078063708037035, −4.18211833303512092634227166706, −3.64694614704696204880925619742, −3.38756120392712418852395697780, −2.94938555094551187344708115777, −1.72711042807289924545237413814, −1.22494467818441012398716771442, 1.22494467818441012398716771442, 1.72711042807289924545237413814, 2.94938555094551187344708115777, 3.38756120392712418852395697780, 3.64694614704696204880925619742, 4.18211833303512092634227166706, 4.39647740261656078063708037035, 5.15060879542347511279056135005, 5.59579701489961261337259153152, 6.03885496050121979241949048776, 6.52686038307702376327906944098, 7.10612424694351373910820587978, 7.53059444797042939960776823323, 7.56844212499678880318083819740, 8.380450659643299525304038593209, 8.391963279391228760465524712718, 9.063305693524270150017827572523, 9.335569883374458015250617223554, 10.27110015600269272082278324988, 10.46615755009783933935676078390

Graph of the ZZ-function along the critical line