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Properties

Label 4-504e2-1.1-c1e2-0-47
Degree 44
Conductor 254016254016
Sign −1-1
Analytic cond. 16.196216.1962
Root an. cond. 2.006102.00610
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  − 2-s − 4-s + 2·7-s + 3·8-s − 2·14-s − 16-s − 10·17-s + 6·23-s + 8·25-s − 2·28-s − 10·31-s − 5·32-s + 10·34-s − 4·41-s − 6·46-s − 6·47-s − 3·49-s − 8·50-s + 6·56-s + 10·62-s + 7·64-s + 10·68-s − 8·71-s − 4·73-s + 6·79-s + 4·82-s + 6·89-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 0.534·14-s − 1/4·16-s − 2.42·17-s + 1.25·23-s + 8/5·25-s − 0.377·28-s − 1.79·31-s − 0.883·32-s + 1.71·34-s − 0.624·41-s − 0.884·46-s − 0.875·47-s − 3/7·49-s − 1.13·50-s + 0.801·56-s + 1.27·62-s + 7/8·64-s + 1.21·68-s − 0.949·71-s − 0.468·73-s + 0.675·79-s + 0.441·82-s + 0.635·89-s + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 16.196216.1962
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 254016, ( :1/2,1/2), −1)(4,\ 254016,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+T+pT2 1 + T + p T^{2}
3 1 1
7C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good5C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
29C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
53C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
59C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
61C22C_2^2 1−12T2+p2T4 1 - 12 T^{2} + p^{2} T^{4}
67C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−14T+pT2)(1+8T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+18T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 18 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

−8.871415429730867931626019060088, −8.310836747793656367534705861594, −7.981273041406930648833538630199, −7.22967943932231695135318750774, −6.93225845707495568453488322983, −6.58582320781888110701679481137, −5.71203585560334225029412443552, −5.01361926234580575550834509635, −4.83452918732394601916818109792, −4.31423602400404044304388677178, −3.63333331448173413922530583590, −2.80952555503907413890783276275, −1.98154620057462151554929555738, −1.30433199199338668521553453852, 0, 1.30433199199338668521553453852, 1.98154620057462151554929555738, 2.80952555503907413890783276275, 3.63333331448173413922530583590, 4.31423602400404044304388677178, 4.83452918732394601916818109792, 5.01361926234580575550834509635, 5.71203585560334225029412443552, 6.58582320781888110701679481137, 6.93225845707495568453488322983, 7.22967943932231695135318750774, 7.981273041406930648833538630199, 8.310836747793656367534705861594, 8.871415429730867931626019060088

Graph of the ZZ-function along the critical line