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Properties

Label 4-160e2-1.1-c5e2-0-1
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 658.508658.508
Root an. cond. 5.065705.06570
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  + 8·3-s + 50·5-s + 104·7-s − 158·9-s + 320·11-s − 100·13-s + 400·15-s + 580·17-s + 720·19-s + 832·21-s + 1.68e3·23-s + 1.87e3·25-s − 1.09e3·27-s + 108·29-s + 9.84e3·31-s + 2.56e3·33-s + 5.20e3·35-s + 6.54e3·37-s − 800·39-s − 1.06e4·41-s + 2.56e4·43-s − 7.90e3·45-s + 2.82e4·47-s − 2.52e4·49-s + 4.64e3·51-s + 3.13e4·53-s + 1.60e4·55-s + ⋯
L(s)  = 1  + 0.513·3-s + 0.894·5-s + 0.802·7-s − 0.650·9-s + 0.797·11-s − 0.164·13-s + 0.459·15-s + 0.486·17-s + 0.457·19-s + 0.411·21-s + 0.665·23-s + 3/5·25-s − 0.289·27-s + 0.0238·29-s + 1.83·31-s + 0.409·33-s + 0.717·35-s + 0.785·37-s − 0.0842·39-s − 0.986·41-s + 2.11·43-s − 0.581·45-s + 1.86·47-s − 1.50·49-s + 0.249·51-s + 1.53·53-s + 0.713·55-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 658.508658.508
Root analytic conductor: 5.065705.06570
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25600, ( :5/2,5/2), 1)(4,\ 25600,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.5441463375.544146337
L(12)L(\frac12) ≈\approx 5.5441463375.544146337
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
5C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
good3D4D_{4} 1−8T+74pT2−8p5T3+p10T4 1 - 8 T + 74 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4}
7D4D_{4} 1−104T+36038T2−104p5T3+p10T4 1 - 104 T + 36038 T^{2} - 104 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1−320T+319702T2−320p5T3+p10T4 1 - 320 T + 319702 T^{2} - 320 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+100T+297086T2+100p5T3+p10T4 1 + 100 T + 297086 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−580T+2475814T2−580p5T3+p10T4 1 - 580 T + 2475814 T^{2} - 580 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1−720T+4633798T2−720p5T3+p10T4 1 - 720 T + 4633798 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1−1688T+10950502T2−1688p5T3+p10T4 1 - 1688 T + 10950502 T^{2} - 1688 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1−108T+39233214T2−108p5T3+p10T4 1 - 108 T + 39233214 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1−9840T+76732702T2−9840p5T3+p10T4 1 - 9840 T + 76732702 T^{2} - 9840 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−6540T+61572814T2−6540p5T3+p10T4 1 - 6540 T + 61572814 T^{2} - 6540 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+10620T+77106582T2+10620p5T3+p10T4 1 + 10620 T + 77106582 T^{2} + 10620 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1−25672T+364912302T2−25672p5T3+p10T4 1 - 25672 T + 364912302 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−28296T+617782998T2−28296p5T3+p10T4 1 - 28296 T + 617782998 T^{2} - 28296 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1−31340T+755347886T2−31340p5T3+p10T4 1 - 31340 T + 755347886 T^{2} - 31340 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1−30800T+1666896598T2−30800p5T3+p10T4 1 - 30800 T + 1666896598 T^{2} - 30800 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1−24540T+1447225822T2−24540p5T3+p10T4 1 - 24540 T + 1447225822 T^{2} - 24540 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1−34584T+2930656478T2−34584p5T3+p10T4 1 - 34584 T + 2930656478 T^{2} - 34584 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+12400T+1143670702T2+12400p5T3+p10T4 1 + 12400 T + 1143670702 T^{2} + 12400 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+7180T+284279286T2+7180p5T3+p10T4 1 + 7180 T + 284279286 T^{2} + 7180 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1−71840T+7430807198T2−71840p5T3+p10T4 1 - 71840 T + 7430807198 T^{2} - 71840 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+31928T+5910993662T2+31928p5T3+p10T4 1 + 31928 T + 5910993662 T^{2} + 31928 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+40748T+8570866774T2+40748p5T3+p10T4 1 + 40748 T + 8570866774 T^{2} + 40748 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+190140T+19653817414T2+190140p5T3+p10T4 1 + 190140 T + 19653817414 T^{2} + 190140 p^{5} T^{3} + 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

−12.10334818284488564526303324012, −11.82704543552943724380179264500, −11.18514327052685527468570444557, −10.81809875389543889540661304757, −10.01484352254298780920174565565, −9.769666204051376942254857200471, −9.049793603702300003964348872616, −8.762616218540841382037509222076, −8.121813769532523070459085255303, −7.70257791294607010306642768406, −6.83374691369088593480486999847, −6.47588285661736547919682466653, −5.49164078420755818348702884741, −5.40962893907297669090338215654, −4.42874427049231063031080235284, −3.81535126494829123129434427721, −2.70963162136766918405162495655, −2.50883897290513374612480264381, −1.34742039953208690732897125716, −0.847092076456144741237769181393, 0.847092076456144741237769181393, 1.34742039953208690732897125716, 2.50883897290513374612480264381, 2.70963162136766918405162495655, 3.81535126494829123129434427721, 4.42874427049231063031080235284, 5.40962893907297669090338215654, 5.49164078420755818348702884741, 6.47588285661736547919682466653, 6.83374691369088593480486999847, 7.70257791294607010306642768406, 8.121813769532523070459085255303, 8.762616218540841382037509222076, 9.049793603702300003964348872616, 9.769666204051376942254857200471, 10.01484352254298780920174565565, 10.81809875389543889540661304757, 11.18514327052685527468570444557, 11.82704543552943724380179264500, 12.10334818284488564526303324012

Graph of the ZZ-function along the critical line