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Properties

Label 8-2160e4-1.1-c1e4-0-3
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
Motivic weight 11
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 − 8·11-s − 4·19-s + 8·25-s − 8·29-s − 24·31-s + 8·41-s + 26·49-s + 32·55-s − 4·61-s − 56·71-s + 4·79-s − 48·89-s + 16·95-s + 48·101-s + 8·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 96·155-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.41·11-s − 0.917·19-s + 8/5·25-s − 1.48·29-s − 4.31·31-s + 1.24·41-s + 26/7·49-s + 4.31·55-s − 0.512·61-s − 6.64·71-s + 0.450·79-s − 5.08·89-s + 1.64·95-s + 4.77·101-s + 8/11·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 7.71·155-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.088873843410.08887384341
L(12)L(\frac12) ≈\approx 0.088873843410.08887384341
L(32)L(\frac{3}{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
5C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
good7C22C_2^2 (1−13T2+p2T4)2 ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}
11D4D_{4} (1+4T+20T2+4pT3+p2T4)2 ( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−2T2+243T4−2p2T6+p4T8 1 - 2 T^{2} + 243 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
19D4D_{4} (1+2T+15T2+2pT3+p2T4)2 ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−72T2+2258T4−72p2T6+p4T8 1 - 72 T^{2} + 2258 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8}
29D4D_{4} (1+4T+56T2+4pT3+p2T4)2 ( 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
31C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
37D4×C2D_4\times C_2 1−50T2+963T4−50p2T6+p4T8 1 - 50 T^{2} + 963 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1−4T+32T2−4pT3+p2T4)2 ( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−192T2+14738T4−192p2T6+p4T8 1 - 192 T^{2} + 14738 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8}
59C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
61D4D_{4} (1+2T+27T2+2pT3+p2T4)2 ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−26T2−453T4−26p2T6+p4T8 1 - 26 T^{2} - 453 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1+28T+332T2+28pT3+p2T4)2 ( 1 + 28 T + 332 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−242T2+25203T4−242p2T6+p4T8 1 - 242 T^{2} + 25203 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1−2T+135T2−2pT3+p2T4)2 ( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−120T2+14978T4−120p2T6+p4T8 1 - 120 T^{2} + 14978 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8}
89C2C_2 (1+12T+pT2)4 ( 1 + 12 T + p T^{2} )^{4}
97D4×C2D_4\times C_2 1−98T2+99pT4−98p2T6+p4T8 1 - 98 T^{2} + 99 p T^{4} - 98 p^{2} T^{6} + p^{4} T^{8}
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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

−6.51048289575479300203341975049, −6.08182364881705438703742063630, −5.78284545154088010144170138681, −5.77040376592785395845103441987, −5.68536610414456997568408830304, −5.48780814344310278778801954649, −5.30897969404684815147865802485, −4.82770065343566124148986431682, −4.71506721794908934577307621314, −4.60076820918027552100909942670, −4.16654553442556505578832680225, −4.07626008454443158242795066397, −3.79399222196726136481918294377, −3.68235378418255673202224734276, −3.60683958366166633862491823643, −3.01177601514116103488112399987, −2.77856740344431491155131005251, −2.71907564912782531246286709430, −2.52729114742290216745072800328, −2.02776601964887935356995357203, −1.79727515288026889875044371097, −1.52776358284870879331170726102, −1.07884395120755782456859970618, −0.33262334093927454019332824298, −0.11894175671256023946648420570, 0.11894175671256023946648420570, 0.33262334093927454019332824298, 1.07884395120755782456859970618, 1.52776358284870879331170726102, 1.79727515288026889875044371097, 2.02776601964887935356995357203, 2.52729114742290216745072800328, 2.71907564912782531246286709430, 2.77856740344431491155131005251, 3.01177601514116103488112399987, 3.60683958366166633862491823643, 3.68235378418255673202224734276, 3.79399222196726136481918294377, 4.07626008454443158242795066397, 4.16654553442556505578832680225, 4.60076820918027552100909942670, 4.71506721794908934577307621314, 4.82770065343566124148986431682, 5.30897969404684815147865802485, 5.48780814344310278778801954649, 5.68536610414456997568408830304, 5.77040376592785395845103441987, 5.78284545154088010144170138681, 6.08182364881705438703742063630, 6.51048289575479300203341975049

Graph of the ZZ-function along the critical line