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Properties

Label 4-70e4-1.1-c1e2-0-2
Degree 44
Conductor 2401000024010000
Sign 11
Analytic cond. 1530.891530.89
Root an. cond. 6.255136.25513
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  − 3·9-s − 4·11-s − 6·29-s + 4·31-s + 10·41-s + 16·59-s + 14·61-s + 16·71-s − 8·79-s − 26·89-s + 12·99-s − 6·101-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s − 1.20·11-s − 1.11·29-s + 0.718·31-s + 1.56·41-s + 2.08·59-s + 1.79·61-s + 1.89·71-s − 0.900·79-s − 2.75·89-s + 1.20·99-s − 0.597·101-s − 1.72·109-s − 0.909·121-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(24010000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(24010000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2401000024010000    =    24⋅54⋅742^{4} \cdot 5^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 1530.891530.89
Root analytic conductor: 6.255136.25513
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 24010000, ( :1/2,1/2), 1)(4,\ 24010000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1220105171.122010517
L(12)L(\frac12) ≈\approx 1.1220105171.122010517
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
7 1 1
good3C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1+35T2+p2T4 1 + 35 T^{2} + p^{2} T^{4}
29C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
43C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C22C_2^2 1−165T2+p2T4 1 - 165 T^{2} + p^{2} T^{4}
89C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} 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

−8.927606384832135505937107537569, −8.244832434877401667982860938815, −7.75819461750138587573321806343, −7.38036159906017304798779246382, −7.11633872769919272074151889212, −6.48257535512681288714544784869, −6.42463364161927056867041349096, −5.64733386520496298938240457734, −5.61406359241653358984327465588, −5.20094445439128630483935361484, −4.97783158302393580326857325548, −4.24254958085369080754719515782, −3.86902306078908157210361053440, −3.70111564415404097309701575085, −2.83743416889672640116737305572, −2.57398002177700183351243545641, −2.48718570798707084033580842034, −1.68351657563366189079604723233, −1.00397152571148135835919303822, −0.32058553311134252531728694303, 0.32058553311134252531728694303, 1.00397152571148135835919303822, 1.68351657563366189079604723233, 2.48718570798707084033580842034, 2.57398002177700183351243545641, 2.83743416889672640116737305572, 3.70111564415404097309701575085, 3.86902306078908157210361053440, 4.24254958085369080754719515782, 4.97783158302393580326857325548, 5.20094445439128630483935361484, 5.61406359241653358984327465588, 5.64733386520496298938240457734, 6.42463364161927056867041349096, 6.48257535512681288714544784869, 7.11633872769919272074151889212, 7.38036159906017304798779246382, 7.75819461750138587573321806343, 8.244832434877401667982860938815, 8.927606384832135505937107537569

Graph of the ZZ-function along the critical line