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Properties

Label 4-28e4-1.1-c5e2-0-6
Degree 44
Conductor 614656614656
Sign 11
Analytic cond. 15810.715810.7
Root an. cond. 11.213411.2134
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  − 14·3-s − 42·5-s − 146·9-s + 716·11-s + 714·13-s + 588·15-s + 1.34e3·17-s − 1.94e3·19-s + 1.79e3·23-s − 102·25-s + 3.43e3·27-s − 1.20e3·29-s − 6.80e3·31-s − 1.00e4·33-s + 1.46e4·37-s − 9.99e3·39-s − 7.89e3·41-s − 524·43-s + 6.13e3·45-s − 1.83e4·47-s − 1.88e4·51-s + 4.51e4·53-s − 3.00e4·55-s + 2.72e4·57-s + 2.25e4·59-s + 5.28e4·61-s − 2.99e4·65-s + ⋯
L(s)  = 1  − 0.898·3-s − 0.751·5-s − 0.600·9-s + 1.78·11-s + 1.17·13-s + 0.674·15-s + 1.12·17-s − 1.23·19-s + 0.706·23-s − 0.0326·25-s + 0.905·27-s − 0.264·29-s − 1.27·31-s − 1.60·33-s + 1.75·37-s − 1.05·39-s − 0.733·41-s − 0.0432·43-s + 0.451·45-s − 1.21·47-s − 1.01·51-s + 2.20·53-s − 1.34·55-s + 1.11·57-s + 0.844·59-s + 1.81·61-s − 0.880·65-s + ⋯

Functional equation

Λ(s)=(614656s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(614656s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 614656614656    =    28⋅742^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 15810.715810.7
Root analytic conductor: 11.213411.2134
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 614656, ( :5/2,5/2), 1)(4,\ 614656,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.3802667182.380266718
L(12)L(\frac12) ≈\approx 2.3802667182.380266718
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
7 1 1
good3D4D_{4} 1+14T+38p2T2+14p5T3+p10T4 1 + 14 T + 38 p^{2} T^{2} + 14 p^{5} T^{3} + p^{10} T^{4}
5D4D_{4} 1+42T+1866T2+42p5T3+p10T4 1 + 42 T + 1866 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1−716T+412438T2−716p5T3+p10T4 1 - 716 T + 412438 T^{2} - 716 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1−714T+814258T2−714p5T3+p10T4 1 - 714 T + 814258 T^{2} - 714 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−1344T+2728510T2−1344p5T3+p10T4 1 - 1344 T + 2728510 T^{2} - 1344 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1+1946T+4229670T2+1946p5T3+p10T4 1 + 1946 T + 4229670 T^{2} + 1946 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1−1792T+11254510T2−1792p5T3+p10T4 1 - 1792 T + 11254510 T^{2} - 1792 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+1200T−16154090T2+1200p5T3+p10T4 1 + 1200 T - 16154090 T^{2} + 1200 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+6804T+30441118T2+6804p5T3+p10T4 1 + 6804 T + 30441118 T^{2} + 6804 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−14640T+187693126T2−14640p5T3+p10T4 1 - 14640 T + 187693126 T^{2} - 14640 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+7896T+209593854T2+7896p5T3+p10T4 1 + 7896 T + 209593854 T^{2} + 7896 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+524T+242298998T2+524p5T3+p10T4 1 + 524 T + 242298998 T^{2} + 524 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+18396T+435885630T2+18396p5T3+p10T4 1 + 18396 T + 435885630 T^{2} + 18396 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1−45132T+1149514990T2−45132p5T3+p10T4 1 - 45132 T + 1149514990 T^{2} - 45132 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1−22582T+1531622854T2−22582p5T3+p10T4 1 - 22582 T + 1531622854 T^{2} - 22582 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1−52822T+2145929546T2−52822p5T3+p10T4 1 - 52822 T + 2145929546 T^{2} - 52822 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+9848T+2714812022T2+9848p5T3+p10T4 1 + 9848 T + 2714812022 T^{2} + 9848 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−840T+427300302T2−840p5T3+p10T4 1 - 840 T + 427300302 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1−122052T+7590539974T2−122052p5T3+p10T4 1 - 122052 T + 7590539974 T^{2} - 122052 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+31704T+6042098590T2+31704p5T3+p10T4 1 + 31704 T + 6042098590 T^{2} + 31704 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1−36974T+4839605030T2−36974p5T3+p10T4 1 - 36974 T + 4839605030 T^{2} - 36974 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1−210588T+21813719542T2−210588p5T3+p10T4 1 - 210588 T + 21813719542 T^{2} - 210588 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1−44240T+2438219582T2−44240p5T3+p10T4 1 - 44240 T + 2438219582 T^{2} - 44240 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

−9.578982954577354140998459930130, −9.451829956363657024653924548405, −8.693338469435825687156315515797, −8.617672310747782743644422657205, −8.099598967670141187943732271770, −7.65982384080178357680129619593, −6.92575044714805092815355437719, −6.72950774177481179857760282477, −6.14474069399899338652475443250, −5.99238098266184798073509570135, −5.18186134089938980220879483256, −5.14689426228218125915424948791, −4.03419559650972499618254299015, −3.89987593956695652274807072993, −3.61453825582208525683394019018, −2.83221284806293859105693043651, −2.07029295629872772380746580973, −1.39365303183960622037605298210, −0.77119680208208657256980920474, −0.48807134220357997457458023684, 0.48807134220357997457458023684, 0.77119680208208657256980920474, 1.39365303183960622037605298210, 2.07029295629872772380746580973, 2.83221284806293859105693043651, 3.61453825582208525683394019018, 3.89987593956695652274807072993, 4.03419559650972499618254299015, 5.14689426228218125915424948791, 5.18186134089938980220879483256, 5.99238098266184798073509570135, 6.14474069399899338652475443250, 6.72950774177481179857760282477, 6.92575044714805092815355437719, 7.65982384080178357680129619593, 8.099598967670141187943732271770, 8.617672310747782743644422657205, 8.693338469435825687156315515797, 9.451829956363657024653924548405, 9.578982954577354140998459930130

Graph of the ZZ-function along the critical line