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Properties

Label 4-20e4-1.1-c5e2-0-3
Degree 44
Conductor 160000160000
Sign 11
Analytic cond. 4115.674115.67
Root an. cond. 8.009588.00958
Motivic weight 55
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  + 342·9-s − 1.08e3·11-s + 1.67e3·19-s + 1.18e3·29-s − 8.51e3·31-s + 3.44e4·41-s + 2.58e4·49-s − 1.53e4·59-s − 6.94e4·61-s + 9.37e4·71-s − 1.53e5·79-s + 5.79e4·81-s − 5.95e4·89-s − 3.69e5·99-s + 2.25e4·101-s − 1.99e5·109-s + 5.52e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.67e5·169-s + ⋯
L(s)  = 1  + 1.40·9-s − 2.69·11-s + 1.06·19-s + 0.262·29-s − 1.59·31-s + 3.20·41-s + 1.53·49-s − 0.573·59-s − 2.39·61-s + 2.20·71-s − 2.77·79-s + 0.980·81-s − 0.796·89-s − 3.78·99-s + 0.220·101-s − 1.61·109-s + 3.43·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.52·169-s + ⋯

Functional equation

Λ(s)=(160000s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(160000s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 160000160000    =    28⋅542^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 4115.674115.67
Root analytic conductor: 8.009588.00958
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 160000, ( :5/2,5/2), 1)(4,\ 160000,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.7967191371.796719137
L(12)L(\frac12) ≈\approx 1.7967191371.796719137
L(72)L(\frac{7}{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
5 1 1
good3C22C_2^2 1−38p2T2+p10T4 1 - 38 p^{2} T^{2} + p^{10} T^{4}
7C22C_2^2 1−25870T2+p10T4 1 - 25870 T^{2} + p^{10} T^{4}
11C2C_2 (1+540T+p5T2)2 ( 1 + 540 T + p^{5} T^{2} )^{2}
13C22C_2^2 1−567862T2+p10T4 1 - 567862 T^{2} + p^{10} T^{4}
17C22C_2^2 1−2486878T2+p10T4 1 - 2486878 T^{2} + p^{10} T^{4}
19C2C_2 (1−44pT+p5T2)2 ( 1 - 44 p T + p^{5} T^{2} )^{2}
23C22C_2^2 1+3970130T2+p10T4 1 + 3970130 T^{2} + p^{10} T^{4}
29C2C_2 (1−594T+p5T2)2 ( 1 - 594 T + p^{5} T^{2} )^{2}
31C2C_2 (1+4256T+p5T2)2 ( 1 + 4256 T + p^{5} T^{2} )^{2}
37C22C_2^2 1−138599110T2+p10T4 1 - 138599110 T^{2} + p^{10} T^{4}
41C2C_2 (1−17226T+p5T2)2 ( 1 - 17226 T + p^{5} T^{2} )^{2}
43C22C_2^2 1−147606886T2+p10T4 1 - 147606886 T^{2} + p^{10} T^{4}
47C22C_2^2 1−457010398T2+p10T4 1 - 457010398 T^{2} + p^{10} T^{4}
53C22C_2^2 1−456374950T2+p10T4 1 - 456374950 T^{2} + p^{10} T^{4}
59C2C_2 (1+7668T+p5T2)2 ( 1 + 7668 T + p^{5} T^{2} )^{2}
61C2C_2 (1+34738T+p5T2)2 ( 1 + 34738 T + p^{5} T^{2} )^{2}
67C22C_2^2 1−2224486870T2+p10T4 1 - 2224486870 T^{2} + p^{10} T^{4}
71C2C_2 (1−46872T+p5T2)2 ( 1 - 46872 T + p^{5} T^{2} )^{2}
73C22C_2^2 1+418480658T2+p10T4 1 + 418480658 T^{2} + p^{10} T^{4}
79C2C_2 (1+76912T+p5T2)2 ( 1 + 76912 T + p^{5} T^{2} )^{2}
83C22C_2^2 1−3292624630T2+p10T4 1 - 3292624630 T^{2} + p^{10} T^{4}
89C2C_2 (1+29754T+p5T2)2 ( 1 + 29754 T + p^{5} T^{2} )^{2}
97C22C_2^2 1−2193410110T2+p10T4 1 - 2193410110 T^{2} + p^{10} T^{4}
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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

−10.62475785789231264987025512885, −10.24305275977156155804027829045, −9.943082018858907392505444495749, −9.190305899352389207720420107799, −9.149797018212453839295726719301, −8.193925878898288663955390245380, −7.73481572594849816144466861808, −7.39088717216677277379639610084, −7.35170991948145031567044906432, −6.43654625103470047892974304877, −5.71146336289234190997462984024, −5.44232128880657713125432632030, −4.92557223846988447624193660257, −4.29457476514301219585111220462, −3.85275774078253835014305425309, −2.81058718284539763051121904714, −2.70282080497535337503635024705, −1.82828414998634702990851181103, −1.10498373110026333324638960889, −0.35694795194284638212864983865, 0.35694795194284638212864983865, 1.10498373110026333324638960889, 1.82828414998634702990851181103, 2.70282080497535337503635024705, 2.81058718284539763051121904714, 3.85275774078253835014305425309, 4.29457476514301219585111220462, 4.92557223846988447624193660257, 5.44232128880657713125432632030, 5.71146336289234190997462984024, 6.43654625103470047892974304877, 7.35170991948145031567044906432, 7.39088717216677277379639610084, 7.73481572594849816144466861808, 8.193925878898288663955390245380, 9.149797018212453839295726719301, 9.190305899352389207720420107799, 9.943082018858907392505444495749, 10.24305275977156155804027829045, 10.62475785789231264987025512885

Graph of the ZZ-function along the critical line