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Properties

Label 4-1134e2-1.1-c1e2-0-1
Degree 44
Conductor 12859561285956
Sign 11
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
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-s − 3·5-s − 7-s − 8-s − 3·10-s − 6·11-s + 13-s − 14-s − 16-s − 6·17-s + 4·19-s − 6·22-s − 6·23-s + 5·25-s + 26-s − 9·29-s + 10·31-s − 6·34-s + 3·35-s − 14·37-s + 4·38-s + 3·40-s + 6·41-s + 4·43-s − 6·46-s + 6·47-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.27·22-s − 1.25·23-s + 25-s + 0.196·26-s − 1.67·29-s + 1.79·31-s − 1.02·34-s + 0.507·35-s − 2.30·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s − 0.884·46-s + 0.875·47-s + 0.707·50-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: 11
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1285956, ( :1/2,1/2), 1)(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.18746515090.1874651509
L(12)L(\frac12) ≈\approx 0.18746515090.1874651509
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3 1 1
7C2C_2 1+T+T2 1 + T + T^{2}
good5C22C_2^2 1+3T+4T2+3pT3+p2T4 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+6T+25T2+6pT3+p2T4 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−T−12T2−pT3+p2T4 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
23C22C_2^2 1+6T+13T2+6pT3+p2T4 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+9T+52T2+9pT3+p2T4 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−10T+69T2−10pT3+p2T4 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4}
37C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
41C22C_2^2 1−6T−5T2−6pT3+p2T4 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−6T−11T2−6pT3+p2T4 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1+6T−23T2+6pT3+p2T4 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+11T+60T2+11pT3+p2T4 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
79C22C_2^2 1+2T−75T2+2pT3+p2T4 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−6T−47T2−6pT3+p2T4 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4}
89C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
97C2C_2 (1−5T+pT2)(1+19T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 19 T + p 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

−10.16521088711055462945433026023, −9.646340327924616780799044735719, −9.062859002339186922990670320013, −8.645155073176911816416612910334, −8.528183196034112520628861756681, −7.70518060214073985202439589059, −7.57574498324002078130841192484, −7.32559103687776293368506055623, −6.68691683329063027033795307091, −6.10538094323615451776469843420, −5.72923805517524127476159205205, −5.29973904621100719966743777549, −4.78143807462183318905549923389, −4.24975329947642706172953511492, −4.01754701711491797000059941657, −3.45912979118964922420805816391, −2.68808397404229597803729587407, −2.65469262308591094433482924622, −1.54331372373995417456064731116, −0.16476133390568287054959784593, 0.16476133390568287054959784593, 1.54331372373995417456064731116, 2.65469262308591094433482924622, 2.68808397404229597803729587407, 3.45912979118964922420805816391, 4.01754701711491797000059941657, 4.24975329947642706172953511492, 4.78143807462183318905549923389, 5.29973904621100719966743777549, 5.72923805517524127476159205205, 6.10538094323615451776469843420, 6.68691683329063027033795307091, 7.32559103687776293368506055623, 7.57574498324002078130841192484, 7.70518060214073985202439589059, 8.528183196034112520628861756681, 8.645155073176911816416612910334, 9.062859002339186922990670320013, 9.646340327924616780799044735719, 10.16521088711055462945433026023

Graph of the ZZ-function along the critical line