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Properties

Label 4-20e4-1.1-c5e2-0-17
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 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 90·9-s − 504·11-s − 440·19-s + 1.38e4·29-s − 1.35e4·31-s − 396·41-s − 3.00e4·49-s − 4.93e4·59-s − 1.13e4·61-s − 1.06e5·71-s + 1.03e5·79-s − 5.09e4·81-s + 1.99e4·89-s + 4.53e4·99-s − 2.18e5·101-s + 4.20e4·109-s − 1.31e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.28e5·169-s + ⋯
L(s)  = 1  − 0.370·9-s − 1.25·11-s − 0.279·19-s + 3.06·29-s − 2.52·31-s − 0.0367·41-s − 1.78·49-s − 1.84·59-s − 0.392·61-s − 2.51·71-s + 1.87·79-s − 0.862·81-s + 0.267·89-s + 0.465·99-s − 2.12·101-s + 0.338·109-s − 0.817·121-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 − 1.96·169-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: 22
Selberg data: (4, 160000, ( :5/2,5/2), 1)(4,\ 160000,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+10p2T2+p10T4 1 + 10 p^{2} T^{2} + p^{10} T^{4}
7C22C_2^2 1+30050T2+p10T4 1 + 30050 T^{2} + p^{10} T^{4}
11C2C_2 (1+252T+p5T2)2 ( 1 + 252 T + p^{5} T^{2} )^{2}
13C22C_2^2 1+728330T2+p10T4 1 + 728330 T^{2} + p^{10} T^{4}
17C22C_2^2 1+2363810T2+p10T4 1 + 2363810 T^{2} + p^{10} T^{4}
19C2C_2 (1+220T+p5T2)2 ( 1 + 220 T + p^{5} T^{2} )^{2}
23C22C_2^2 1+6946370T2+p10T4 1 + 6946370 T^{2} + p^{10} T^{4}
29C2C_2 (1−6930T+p5T2)2 ( 1 - 6930 T + p^{5} T^{2} )^{2}
31C2C_2 (1+6752T+p5T2)2 ( 1 + 6752 T + p^{5} T^{2} )^{2}
37C22C_2^2 1−56462470T2+p10T4 1 - 56462470 T^{2} + p^{10} T^{4}
41C2C_2 (1+198T+p5T2)2 ( 1 + 198 T + p^{5} T^{2} )^{2}
43C22C_2^2 1+293842250T2+p10T4 1 + 293842250 T^{2} + p^{10} T^{4}
47C22C_2^2 1+347593490T2+p10T4 1 + 347593490 T^{2} + p^{10} T^{4}
53C22C_2^2 1+802472090T2+p10T4 1 + 802472090 T^{2} + p^{10} T^{4}
59C2C_2 (1+24660T+p5T2)2 ( 1 + 24660 T + p^{5} T^{2} )^{2}
61C2C_2 (1+5698T+p5T2)2 ( 1 + 5698 T + p^{5} T^{2} )^{2}
67C22C_2^2 1+795787610T2+p10T4 1 + 795787610 T^{2} + p^{10} T^{4}
71C2C_2 (1+53352T+p5T2)2 ( 1 + 53352 T + p^{5} T^{2} )^{2}
73C22C_2^2 1−883886830T2+p10T4 1 - 883886830 T^{2} + p^{10} T^{4}
79C2C_2 (1−51920T+p5T2)2 ( 1 - 51920 T + p^{5} T^{2} )^{2}
83C22C_2^2 1+4053674810T2+p10T4 1 + 4053674810 T^{2} + p^{10} T^{4}
89C2C_2 (1−9990T+p5T2)2 ( 1 - 9990 T + p^{5} T^{2} )^{2}
97C22C_2^2 1+6923133890T2+p10T4 1 + 6923133890 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

−10.35818821787494541279214543999, −9.883887973933525492609155786275, −9.123310556186932417007813270567, −9.024465740310844799764981421558, −8.183758468446020490020128732260, −8.091083777066748536762112707685, −7.47399857116392210165083014456, −6.99548608481181573793524656547, −6.28388660677356580970282665761, −6.07930988280287988379847523773, −5.18607476741954390001413273813, −5.03330167465079049744988522040, −4.39028019376438201796000059209, −3.67262407964577863461859941301, −2.89022068347581338570184162864, −2.72879363999033578777438028677, −1.79199034474100036482378086367, −1.17575047012676916630540584565, 0, 0, 1.17575047012676916630540584565, 1.79199034474100036482378086367, 2.72879363999033578777438028677, 2.89022068347581338570184162864, 3.67262407964577863461859941301, 4.39028019376438201796000059209, 5.03330167465079049744988522040, 5.18607476741954390001413273813, 6.07930988280287988379847523773, 6.28388660677356580970282665761, 6.99548608481181573793524656547, 7.47399857116392210165083014456, 8.091083777066748536762112707685, 8.183758468446020490020128732260, 9.024465740310844799764981421558, 9.123310556186932417007813270567, 9.883887973933525492609155786275, 10.35818821787494541279214543999

Graph of the ZZ-function along the critical line