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Properties

Label 4-160e2-1.1-c6e2-0-1
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 1354.871354.87
Root an. cond. 6.067016.06701
Motivic weight 66
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 88·5-s + 2.41e3·13-s + 1.07e4·17-s − 7.88e3·25-s + 2.92e4·37-s + 2.17e5·41-s + 2.66e5·53-s + 7.76e5·61-s + 2.12e5·65-s + 1.07e6·73-s − 5.31e5·81-s + 9.47e5·85-s − 3.93e5·97-s + 3.40e6·101-s − 3.78e6·113-s − 3.54e6·121-s − 2.06e6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.91e6·169-s + ⋯
L(s)  = 1  + 0.703·5-s + 1.09·13-s + 2.19·17-s − 0.504·25-s + 0.577·37-s + 3.15·41-s + 1.79·53-s + 3.42·61-s + 0.773·65-s + 2.77·73-s − 81-s + 1.54·85-s − 0.431·97-s + 3.30·101-s − 2.62·113-s − 2·121-s − 1.05·125-s + 0.603·169-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+3)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 1354.871354.87
Root analytic conductor: 6.067016.06701
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25600, ( :3,3), 1)(4,\ 25600,\ (\ :3, 3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 5.0625365095.062536509
L(12)L(\frac12) ≈\approx 5.0625365095.062536509
L(4)L(4) 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
5C2C_2 1−88T+p6T2 1 - 88 T + p^{6} T^{2}
good3C22C_2^2 1+p12T4 1 + p^{12} T^{4}
7C22C_2^2 1+p12T4 1 + p^{12} T^{4}
11C2C_2 (1+p6T2)2 ( 1 + p^{6} T^{2} )^{2}
13C2C_2 (1−4070T+p6T2)(1+1656T+p6T2) ( 1 - 4070 T + p^{6} T^{2} )( 1 + 1656 T + p^{6} T^{2} )
17C2C_2 (1−9776T+p6T2)(1−990T+p6T2) ( 1 - 9776 T + p^{6} T^{2} )( 1 - 990 T + p^{6} T^{2} )
19C1C_1×\timesC1C_1 (1−p3T)2(1+p3T)2 ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2}
23C22C_2^2 1+p12T4 1 + p^{12} T^{4}
29C2C_2 (1−31878T+p6T2)(1+31878T+p6T2) ( 1 - 31878 T + p^{6} T^{2} )( 1 + 31878 T + p^{6} T^{2} )
31C2C_2 (1+p6T2)2 ( 1 + p^{6} T^{2} )^{2}
37C2C_2 (1−84744T+p6T2)(1+55510T+p6T2) ( 1 - 84744 T + p^{6} T^{2} )( 1 + 55510 T + p^{6} T^{2} )
41C2C_2 (1−108560T+p6T2)2 ( 1 - 108560 T + p^{6} T^{2} )^{2}
43C22C_2^2 1+p12T4 1 + p^{12} T^{4}
47C22C_2^2 1+p12T4 1 + p^{12} T^{4}
53C2C_2 (1−296296T+p6T2)(1+29430T+p6T2) ( 1 - 296296 T + p^{6} T^{2} )( 1 + 29430 T + p^{6} T^{2} )
59C1C_1×\timesC1C_1 (1−p3T)2(1+p3T)2 ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2}
61C2C_2 (1−388440T+p6T2)2 ( 1 - 388440 T + p^{6} T^{2} )^{2}
67C22C_2^2 1+p12T4 1 + p^{12} T^{4}
71C2C_2 (1+p6T2)2 ( 1 + p^{6} T^{2} )^{2}
73C2C_2 (1−650016T+p6T2)(1−427570T+p6T2) ( 1 - 650016 T + p^{6} T^{2} )( 1 - 427570 T + p^{6} T^{2} )
79C1C_1×\timesC1C_1 (1−p3T)2(1+p3T)2 ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2}
83C22C_2^2 1+p12T4 1 + p^{12} T^{4}
89C2C_2 (1−1378962T+p6T2)(1+1378962T+p6T2) ( 1 - 1378962 T + p^{6} T^{2} )( 1 + 1378962 T + p^{6} T^{2} )
97C2C_2 (1−1078704T+p6T2)(1+1472510T+p6T2) ( 1 - 1078704 T + p^{6} T^{2} )( 1 + 1472510 T + p^{6} 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

−11.77826147724802193678895508889, −11.74903926950078118159650607705, −10.90651481536231729961307332732, −10.55828768908469105864198692976, −9.791716052827979083852667487119, −9.725819439488743672562265660110, −9.012873831689916744092270131356, −8.412016659401069092661961509591, −7.81898226038400046764804939245, −7.47050056225287782711781843014, −6.59675133087085575457141375008, −6.05947591637361310994492279519, −5.54478879738374444127527691169, −5.21273802763326736988995812034, −3.91723425199992899462021158853, −3.82596384975116021717640824623, −2.74794547442634824250810588265, −2.15379884661766649270034891376, −1.05311479467472686699189203650, −0.841774997991425983080826553616, 0.841774997991425983080826553616, 1.05311479467472686699189203650, 2.15379884661766649270034891376, 2.74794547442634824250810588265, 3.82596384975116021717640824623, 3.91723425199992899462021158853, 5.21273802763326736988995812034, 5.54478879738374444127527691169, 6.05947591637361310994492279519, 6.59675133087085575457141375008, 7.47050056225287782711781843014, 7.81898226038400046764804939245, 8.412016659401069092661961509591, 9.012873831689916744092270131356, 9.725819439488743672562265660110, 9.791716052827979083852667487119, 10.55828768908469105864198692976, 10.90651481536231729961307332732, 11.74903926950078118159650607705, 11.77826147724802193678895508889

Graph of the ZZ-function along the critical line