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Properties

Label 4-504e2-1.1-c5e2-0-5
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 6534.046534.04
Root an. cond. 8.990748.99074
Motivic weight 55
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  + 62·5-s + 98·7-s − 972·11-s + 78·13-s − 560·17-s + 2.64e3·19-s − 2.27e3·23-s + 1.05e3·25-s + 7.80e3·29-s + 5.44e3·31-s + 6.07e3·35-s + 576·37-s + 1.68e4·41-s − 8.39e3·43-s + 4.53e3·47-s + 7.20e3·49-s − 1.42e3·53-s − 6.02e4·55-s − 3.41e4·59-s + 1.91e4·61-s + 4.83e3·65-s + 5.69e4·67-s + 7.22e3·71-s + 1.28e5·73-s − 9.52e4·77-s + 5.28e4·79-s + 8.44e4·83-s + ⋯
L(s)  = 1  + 1.10·5-s + 0.755·7-s − 2.42·11-s + 0.128·13-s − 0.469·17-s + 1.67·19-s − 0.895·23-s + 0.338·25-s + 1.72·29-s + 1.01·31-s + 0.838·35-s + 0.0691·37-s + 1.56·41-s − 0.692·43-s + 0.299·47-s + 3/7·49-s − 0.0694·53-s − 2.68·55-s − 1.27·59-s + 0.657·61-s + 0.141·65-s + 1.54·67-s + 0.170·71-s + 2.82·73-s − 1.83·77-s + 0.951·79-s + 1.34·83-s + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 6534.046534.04
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 254016, ( :5/2,5/2), 1)(4,\ 254016,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.9671022584.967102258
L(12)L(\frac12) ≈\approx 4.9671022584.967102258
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
3 1 1
7C1C_1 (1−p2T)2 ( 1 - p^{2} T )^{2}
good5D4D_{4} 1−62T+2786T2−62p5T3+p10T4 1 - 62 T + 2786 T^{2} - 62 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+972T+551926T2+972p5T3+p10T4 1 + 972 T + 551926 T^{2} + 972 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1−6pT−148150T2−6p6T3+p10T4 1 - 6 p T - 148150 T^{2} - 6 p^{6} T^{3} + p^{10} T^{4}
17D4D_{4} 1+560T+2911742T2+560p5T3+p10T4 1 + 560 T + 2911742 T^{2} + 560 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1−2642T+6454926T2−2642p5T3+p10T4 1 - 2642 T + 6454926 T^{2} - 2642 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+2272T+3276974T2+2272p5T3+p10T4 1 + 2272 T + 3276974 T^{2} + 2272 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1−7808T+56228822T2−7808p5T3+p10T4 1 - 7808 T + 56228822 T^{2} - 7808 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1−5444T+61676286T2−5444p5T3+p10T4 1 - 5444 T + 61676286 T^{2} - 5444 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−576T+63988358T2−576p5T3+p10T4 1 - 576 T + 63988358 T^{2} - 576 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1−16888T+277723070T2−16888p5T3+p10T4 1 - 16888 T + 277723070 T^{2} - 16888 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+8396T+291418902T2+8396p5T3+p10T4 1 + 8396 T + 291418902 T^{2} + 8396 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−4532T+192546782T2−4532p5T3+p10T4 1 - 4532 T + 192546782 T^{2} - 4532 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+1420T+601997678T2+1420p5T3+p10T4 1 + 1420 T + 601997678 T^{2} + 1420 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+34146T+1702640302T2+34146p5T3+p10T4 1 + 34146 T + 1702640302 T^{2} + 34146 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1−19106T−202373982T2−19106p5T3+p10T4 1 - 19106 T - 202373982 T^{2} - 19106 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1−56952T+3238977590T2−56952p5T3+p10T4 1 - 56952 T + 3238977590 T^{2} - 56952 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−7224T+3534775246T2−7224p5T3+p10T4 1 - 7224 T + 3534775246 T^{2} - 7224 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1−128828T+8014301382T2−128828p5T3+p10T4 1 - 128828 T + 8014301382 T^{2} - 128828 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1−52808T+6427138206T2−52808p5T3+p10T4 1 - 52808 T + 6427138206 T^{2} - 52808 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1−84486T+4859890798T2−84486p5T3+p10T4 1 - 84486 T + 4859890798 T^{2} - 84486 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1−130972T+14510409206T2−130972p5T3+p10T4 1 - 130972 T + 14510409206 T^{2} - 130972 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1−194624T+26622623358T2−194624p5T3+p10T4 1 - 194624 T + 26622623358 T^{2} - 194624 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

−10.36642534538784457901632765727, −9.898991370242585629651535132176, −9.591656256696332292364270270137, −9.138289492770311183793425115355, −8.388355494805211027420834492546, −8.012821283629397858902762720153, −7.83270108465493667520821741688, −7.33852326907518346972394024383, −6.38970651874797197063000701809, −6.36233767872178641266434230549, −5.47788046569421298790724544548, −5.24261944571308483278222942633, −4.90453838959611915338473952714, −4.32142900050270701478367841349, −3.37962327418072056594856622378, −2.84273132491704866433304713472, −2.20794676708162849652633382074, −2.05373339389832616561133157460, −0.897039847065639093826117081168, −0.63194354444668070666829272954, 0.63194354444668070666829272954, 0.897039847065639093826117081168, 2.05373339389832616561133157460, 2.20794676708162849652633382074, 2.84273132491704866433304713472, 3.37962327418072056594856622378, 4.32142900050270701478367841349, 4.90453838959611915338473952714, 5.24261944571308483278222942633, 5.47788046569421298790724544548, 6.36233767872178641266434230549, 6.38970651874797197063000701809, 7.33852326907518346972394024383, 7.83270108465493667520821741688, 8.012821283629397858902762720153, 8.388355494805211027420834492546, 9.138289492770311183793425115355, 9.591656256696332292364270270137, 9.898991370242585629651535132176, 10.36642534538784457901632765727

Graph of the ZZ-function along the critical line