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Properties

Label 4-1280e2-1.1-c1e2-0-16
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
Motivic weight 11
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  + 2·5-s − 4·9-s − 4·13-s + 12·17-s + 3·25-s − 8·29-s + 4·37-s + 16·41-s − 8·45-s + 4·49-s − 4·53-s − 28·61-s − 8·65-s + 12·73-s + 7·81-s + 24·85-s + 12·89-s + 20·97-s + 12·109-s − 20·113-s + 16·117-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 4/3·9-s − 1.10·13-s + 2.91·17-s + 3/5·25-s − 1.48·29-s + 0.657·37-s + 2.49·41-s − 1.19·45-s + 4/7·49-s − 0.549·53-s − 3.58·61-s − 0.992·65-s + 1.40·73-s + 7/9·81-s + 2.60·85-s + 1.27·89-s + 2.03·97-s + 1.14·109-s − 1.88·113-s + 1.47·117-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2069340322.206934032
L(12)L(\frac12) ≈\approx 2.2069340322.206934032
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)
bad2 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
7C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
23C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
29C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
31C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
43C22C_2^2 1+84T2+p2T4 1 + 84 T^{2} + p^{2} T^{4}
47C22C_2^2 1+92T2+p2T4 1 + 92 T^{2} + p^{2} T^{4}
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C22C_2^2 1+116T2+p2T4 1 + 116 T^{2} + p^{2} T^{4}
71C22C_2^2 1+134T2+p2T4 1 + 134 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
83C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{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

−9.761164842599230452270933274350, −9.417813960759149650388470402767, −9.146917843984336564199460927881, −8.934543866025143240598883397938, −7.965800338953623128029847401656, −7.84804635143844064735293788203, −7.60923321559064389382275585544, −7.16521819535438178940961557869, −6.26415304508219311984899254644, −6.11974118255946195368307126920, −5.58415167840856839644784371779, −5.49936600934448183561518176378, −4.90639145286840195823956743878, −4.40058944195331033734735040382, −3.58082100235383699110499102578, −3.20684118995595838914477826137, −2.72144630647887323038483034788, −2.22259305792153845108702480225, −1.45358908317161389346517381853, −0.64110331270867126694107758633, 0.64110331270867126694107758633, 1.45358908317161389346517381853, 2.22259305792153845108702480225, 2.72144630647887323038483034788, 3.20684118995595838914477826137, 3.58082100235383699110499102578, 4.40058944195331033734735040382, 4.90639145286840195823956743878, 5.49936600934448183561518176378, 5.58415167840856839644784371779, 6.11974118255946195368307126920, 6.26415304508219311984899254644, 7.16521819535438178940961557869, 7.60923321559064389382275585544, 7.84804635143844064735293788203, 7.965800338953623128029847401656, 8.934543866025143240598883397938, 9.146917843984336564199460927881, 9.417813960759149650388470402767, 9.761164842599230452270933274350

Graph of the ZZ-function along the critical line