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Properties

Label 4-40e4-1.1-c1e2-0-3
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
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  − 2·13-s − 6·17-s − 14·37-s − 16·41-s + 18·53-s − 24·61-s + 22·73-s − 9·81-s − 26·97-s − 4·101-s + 2·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 0.554·13-s − 1.45·17-s − 2.30·37-s − 2.49·41-s + 2.47·53-s − 3.07·61-s + 2.57·73-s − 81-s − 2.63·97-s − 0.398·101-s + 0.188·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.91954277210.9195427721
L(12)L(\frac12) ≈\approx 0.91954277210.9195427721
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
5 1 1
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
7C22C_2^2 1+p2T4 1 + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1+p2T4 1 + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1+2T+pT2)(1+12T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
43C22C_2^2 1+p2T4 1 + p^{2} T^{4}
47C22C_2^2 1+p2T4 1 + p^{2} T^{4}
53C2C_2 (1−14T+pT2)(1−4T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
67C22C_2^2 1+p2T4 1 + p^{2} T^{4}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1−6T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+p2T4 1 + p^{2} T^{4}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1+8T+pT2)(1+18T+pT2) ( 1 + 8 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

−9.767903981935925295708492327207, −9.158047673976675898491476918859, −8.687258232474921851433377090122, −8.505857169049932365334168389233, −8.226234801848473967582035547826, −7.39847823269285495455918527513, −7.25980775975196894870560459596, −6.77304054145758788074451628387, −6.55043374149371581466503007737, −5.95840057996330046255882319759, −5.44955610623167730756439373762, −4.95607936423897530364484505212, −4.81645057521060485154422430691, −4.04574733067508784466632777220, −3.76485928125883358608548295484, −3.10195229467936784608381047703, −2.62941979123255925439395993849, −1.92297272797417177597877392857, −1.58776063976500662580433775881, −0.35694884930754046654729483871, 0.35694884930754046654729483871, 1.58776063976500662580433775881, 1.92297272797417177597877392857, 2.62941979123255925439395993849, 3.10195229467936784608381047703, 3.76485928125883358608548295484, 4.04574733067508784466632777220, 4.81645057521060485154422430691, 4.95607936423897530364484505212, 5.44955610623167730756439373762, 5.95840057996330046255882319759, 6.55043374149371581466503007737, 6.77304054145758788074451628387, 7.25980775975196894870560459596, 7.39847823269285495455918527513, 8.226234801848473967582035547826, 8.505857169049932365334168389233, 8.687258232474921851433377090122, 9.158047673976675898491476918859, 9.767903981935925295708492327207

Graph of the ZZ-function along the critical line