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Properties

Label 8-336e4-1.1-c2e4-0-5
Degree 88
Conductor 1274550681612745506816
Sign 11
Analytic cond. 7025.827025.82
Root an. cond. 3.025773.02577
Motivic weight 22
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  + 4·5-s − 6·9-s + 32·13-s − 52·17-s − 48·25-s − 8·29-s + 80·37-s + 68·41-s − 24·45-s − 14·49-s + 16·53-s − 264·61-s + 128·65-s − 272·73-s + 27·81-s − 208·85-s − 172·89-s + 320·97-s − 436·101-s − 88·109-s + 424·113-s − 192·117-s + 464·121-s − 228·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 4/5·5-s − 2/3·9-s + 2.46·13-s − 3.05·17-s − 1.91·25-s − 0.275·29-s + 2.16·37-s + 1.65·41-s − 0.533·45-s − 2/7·49-s + 0.301·53-s − 4.32·61-s + 1.96·65-s − 3.72·73-s + 1/3·81-s − 2.44·85-s − 1.93·89-s + 3.29·97-s − 4.31·101-s − 0.807·109-s + 3.75·113-s − 1.64·117-s + 3.83·121-s − 1.82·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

Λ(s)=((216⋅34⋅74)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅34⋅74)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅34⋅742^{16} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 7025.827025.82
Root analytic conductor: 3.025773.02577
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅34⋅74, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.96849664780.9684966478
L(12)L(\frac12) ≈\approx 0.96849664780.9684966478
L(2)L(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
3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
good5D4D_{4} (1−2T+6pT2−2p2T3+p4T4)2 ( 1 - 2 T + 6 p T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
11D4×C2D_4\times C_2 1−464T2+83022T4−464p4T6+p8T8 1 - 464 T^{2} + 83022 T^{4} - 464 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1−16T+318T2−16p2T3+p4T4)2 ( 1 - 16 T + 318 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4D_{4} (1+26T+558T2+26p2T3+p4T4)2 ( 1 + 26 T + 558 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4×C2D_4\times C_2 1−1124T2+554982T4−1124p4T6+p8T8 1 - 1124 T^{2} + 554982 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8}
23D4×C2D_4\times C_2 1−1280T2+866382T4−1280p4T6+p8T8 1 - 1280 T^{2} + 866382 T^{4} - 1280 p^{4} T^{6} + p^{8} T^{8}
29C2C_2 (1+2T+p2T2)4 ( 1 + 2 T + p^{2} T^{2} )^{4}
31D4×C2D_4\times C_2 1−1444T2+1594182T4−1444p4T6+p8T8 1 - 1444 T^{2} + 1594182 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8}
37D4D_{4} (1−40T+2382T2−40p2T3+p4T4)2 ( 1 - 40 T + 2382 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4D_{4} (1−34T+102T2−34p2T3+p4T4)2 ( 1 - 34 T + 102 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4×C2D_4\times C_2 1−2276T2+2627622T4−2276p4T6+p8T8 1 - 2276 T^{2} + 2627622 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8}
47D4×C2D_4\times C_2 1−7156T2+21968742T4−7156p4T6+p8T8 1 - 7156 T^{2} + 21968742 T^{4} - 7156 p^{4} T^{6} + p^{8} T^{8}
53D4D_{4} (1−8T+3534T2−8p2T3+p4T4)2 ( 1 - 8 T + 3534 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2}
59D4×C2D_4\times C_2 1−5204T2+17832582T4−5204p4T6+p8T8 1 - 5204 T^{2} + 17832582 T^{4} - 5204 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+132T+8774T2+132p2T3+p4T4)2 ( 1 + 132 T + 8774 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4×C2D_4\times C_2 1−12244T2+72446790T4−12244p4T6+p8T8 1 - 12244 T^{2} + 72446790 T^{4} - 12244 p^{4} T^{6} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−12704T2+90771342T4−12704p4T6+p8T8 1 - 12704 T^{2} + 90771342 T^{4} - 12704 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+136T+14526T2+136p2T3+p4T4)2 ( 1 + 136 T + 14526 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1−11764T2+106099302T4−11764p4T6+p8T8 1 - 11764 T^{2} + 106099302 T^{4} - 11764 p^{4} T^{6} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−13124T2+100043430T4−13124p4T6+p8T8 1 - 13124 T^{2} + 100043430 T^{4} - 13124 p^{4} T^{6} + p^{8} T^{8}
89D4D_{4} (1+86T+11622T2+86p2T3+p4T4)2 ( 1 + 86 T + 11622 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4D_{4} (1−160T+24462T2−160p2T3+p4T4)2 ( 1 - 160 T + 24462 T^{2} - 160 p^{2} T^{3} + p^{4} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.291268838339024134350217859146, −7.86343455123646010052104887893, −7.82252355433795393305931857835, −7.25341379673887070567247729885, −7.22346200711701779626851718488, −6.92787894064860714349355581566, −6.30223976147776920791094645891, −6.20635598909133884367187035819, −6.17628448713047085910445165886, −5.89818173159035827076170462210, −5.71469711778685439179765640499, −5.57574571497726934448056107360, −4.67097754839930846468383883589, −4.60090305944645756590294390489, −4.49467769692893253723677708308, −4.04301177016299285349902101101, −3.92979426896007412020465047647, −3.30800246523290376398010510346, −3.10883524977804001640267427896, −2.56173708265340695628739762416, −2.42076055408657023679901619994, −1.79319150435058700318023390080, −1.64614872806148064218081681700, −1.07079857837807459949058201527, −0.21708212336815606221865577364, 0.21708212336815606221865577364, 1.07079857837807459949058201527, 1.64614872806148064218081681700, 1.79319150435058700318023390080, 2.42076055408657023679901619994, 2.56173708265340695628739762416, 3.10883524977804001640267427896, 3.30800246523290376398010510346, 3.92979426896007412020465047647, 4.04301177016299285349902101101, 4.49467769692893253723677708308, 4.60090305944645756590294390489, 4.67097754839930846468383883589, 5.57574571497726934448056107360, 5.71469711778685439179765640499, 5.89818173159035827076170462210, 6.17628448713047085910445165886, 6.20635598909133884367187035819, 6.30223976147776920791094645891, 6.92787894064860714349355581566, 7.22346200711701779626851718488, 7.25341379673887070567247729885, 7.82252355433795393305931857835, 7.86343455123646010052104887893, 8.291268838339024134350217859146

Graph of the ZZ-function along the critical line