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Properties

Label 4-350e2-1.1-c5e2-0-8
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 3151.063151.06
Root an. cond. 7.492287.49228
Motivic weight 55
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  − 16·4-s − 43·9-s + 1.11e3·11-s + 256·16-s + 4.07e3·19-s + 1.00e4·29-s + 1.13e4·31-s + 688·36-s − 4.84e3·41-s − 1.77e4·44-s − 2.40e3·49-s − 1.14e4·59-s − 7.22e4·61-s − 4.09e3·64-s + 3.21e4·71-s − 6.52e4·76-s + 1.28e5·79-s − 5.72e4·81-s + 1.43e5·89-s − 4.77e4·99-s − 1.14e5·101-s − 1.76e5·109-s − 1.60e5·116-s + 6.01e5·121-s − 1.82e5·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.176·9-s + 2.76·11-s + 1/4·16-s + 2.59·19-s + 2.20·29-s + 2.12·31-s + 0.0884·36-s − 0.450·41-s − 1.38·44-s − 1/7·49-s − 0.428·59-s − 2.48·61-s − 1/8·64-s + 0.757·71-s − 1.29·76-s + 2.31·79-s − 0.968·81-s + 1.91·89-s − 0.489·99-s − 1.11·101-s − 1.42·109-s − 1.10·116-s + 3.73·121-s − 1.06·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 3151.063151.06
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :5/2,5/2), 1)(4,\ 122500,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.8877614904.887761490
L(12)L(\frac12) ≈\approx 4.8877614904.887761490
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)
bad2C2C_2 1+p4T2 1 + p^{4} T^{2}
5 1 1
7C2C_2 1+p4T2 1 + p^{4} T^{2}
good3C22C_2^2 1+43T2+p10T4 1 + 43 T^{2} + p^{10} T^{4}
11C2C_2 (1−555T+p5T2)2 ( 1 - 555 T + p^{5} T^{2} )^{2}
13C22C_2^2 1−684505T2+p10T4 1 - 684505 T^{2} + p^{10} T^{4}
17C22C_2^2 1−616633T2+p10T4 1 - 616633 T^{2} + p^{10} T^{4}
19C2C_2 (1−2038T+p5T2)2 ( 1 - 2038 T + p^{5} T^{2} )^{2}
23C22C_2^2 1−11359786T2+p10T4 1 - 11359786 T^{2} + p^{10} T^{4}
29C2C_2 (1−5001T+p5T2)2 ( 1 - 5001 T + p^{5} T^{2} )^{2}
31C2C_2 (1−5696T+p5T2)2 ( 1 - 5696 T + p^{5} T^{2} )^{2}
37C22C_2^2 1−107305510T2+p10T4 1 - 107305510 T^{2} + p^{10} T^{4}
41C2C_2 (1+2424T+p5T2)2 ( 1 + 2424 T + p^{5} T^{2} )^{2}
43C22C_2^2 1−158818p2T2+p10T4 1 - 158818 p^{2} T^{2} + p^{10} T^{4}
47C22C_2^2 1+77834555T2+p10T4 1 + 77834555 T^{2} + p^{10} T^{4}
53C22C_2^2 1−196503370T2+p10T4 1 - 196503370 T^{2} + p^{10} T^{4}
59C2C_2 (1+5724T+p5T2)2 ( 1 + 5724 T + p^{5} T^{2} )^{2}
61C2C_2 (1+592pT+p5T2)2 ( 1 + 592 p T + p^{5} T^{2} )^{2}
67C22C_2^2 1+1669488602T2+p10T4 1 + 1669488602 T^{2} + p^{10} T^{4}
71C2C_2 (1−16080T+p5T2)2 ( 1 - 16080 T + p^{5} T^{2} )^{2}
73C22C_2^2 1+2331209138T2+p10T4 1 + 2331209138 T^{2} + p^{10} T^{4}
79C2C_2 (1−64147T+p5T2)2 ( 1 - 64147 T + p^{5} T^{2} )^{2}
83C22C_2^2 1+3418207370T2+p10T4 1 + 3418207370 T^{2} + p^{10} T^{4}
89C2C_2 (1−71676T+p5T2)2 ( 1 - 71676 T + p^{5} T^{2} )^{2}
97C22C_2^2 1+5633870111T2+p10T4 1 + 5633870111 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

−10.79436551992783854040129524485, −10.46105753580863299980992283273, −9.665584866555970450980743718289, −9.456745655161154827255724815166, −9.301828437467604673834132144747, −8.531504222638364019542957352750, −8.221087832401596413136055544246, −7.63559629086931441032039897191, −6.96674019568112547110125440797, −6.49643343210158103303950850906, −6.25026670490442218478676980501, −5.53427091592031866883421927762, −4.66975685954181328755070924381, −4.63320630788029861859570397367, −3.75633852047851529771488045172, −3.28007492211831578059512920654, −2.77116584728048343214059821393, −1.55533415445948253457156174616, −1.06191224531082288781426379242, −0.73413857434771612288531027909, 0.73413857434771612288531027909, 1.06191224531082288781426379242, 1.55533415445948253457156174616, 2.77116584728048343214059821393, 3.28007492211831578059512920654, 3.75633852047851529771488045172, 4.63320630788029861859570397367, 4.66975685954181328755070924381, 5.53427091592031866883421927762, 6.25026670490442218478676980501, 6.49643343210158103303950850906, 6.96674019568112547110125440797, 7.63559629086931441032039897191, 8.221087832401596413136055544246, 8.531504222638364019542957352750, 9.301828437467604673834132144747, 9.456745655161154827255724815166, 9.665584866555970450980743718289, 10.46105753580863299980992283273, 10.79436551992783854040129524485

Graph of the ZZ-function along the critical line