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Properties

Label 128-3288e64-1.1-c0e64-0-1
Degree $128$
Conductor $1.212\times 10^{225}$
Sign $1$
Analytic cond. $5.83031\times 10^{13}$
Root an. cond. $1.28098$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16-s − 4·17-s − 4·19-s + 4·41-s + 81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 16-s − 4·17-s − 4·19-s + 4·41-s + 81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{192} \cdot 3^{64} \cdot 137^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{192} \cdot 3^{64} \cdot 137^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(128\)
Conductor: \(2^{192} \cdot 3^{64} \cdot 137^{64}\)
Sign: $1$
Analytic conductor: \(5.83031\times 10^{13}\)
Root analytic conductor: \(1.28098\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((128,\ 2^{192} \cdot 3^{64} \cdot 137^{64} ,\ ( \ : [0]^{64} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06263581647\)
\(L(\frac12)\) \(\approx\) \(0.06263581647\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} \)
3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} \)
137 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} \)
good5 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
7 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \)
11 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \)
13 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2} \)
23 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
29 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
31 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
37 \( ( 1 + T^{4} )^{32} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
47 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
53 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
71 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} )^{2} \)
79 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} - T^{40} + T^{48} - T^{56} + T^{64} - T^{72} + T^{80} - T^{88} + T^{96} - T^{104} + T^{112} - T^{120} + T^{128} \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
97 \( ( 1 + T^{2} )^{32}( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} - T^{52} + T^{56} - T^{60} + T^{64} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{128} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−0.956712825070201278336939414552, −0.932098302050877510774054579851, −0.910832898679790027680075763417, −0.910731563362321905198238877443, −0.905698032598951047770596246621, −0.897106061832670160802316306977, −0.880475687264510960124882687595, −0.859445927344910753202356439444, −0.817225582934635751764537729036, −0.76280245737677060794066122129, −0.75901491367071960693398626885, −0.74080983354681324338173521789, −0.72482237925929475243270641245, −0.67526653123164133660888894119, −0.67287019176154773191247876942, −0.61055722451858667367914297466, −0.46293893816338460349832548167, −0.45355417259430097810314927631, −0.38383784464807140003542984287, −0.34729226015492456599045824464, −0.34213562329734764708501658637, −0.20608681794878526207030163393, −0.16426625489173871265450119406, −0.12928389972225329223786611698, −0.10366525926397433129272999741, 0.10366525926397433129272999741, 0.12928389972225329223786611698, 0.16426625489173871265450119406, 0.20608681794878526207030163393, 0.34213562329734764708501658637, 0.34729226015492456599045824464, 0.38383784464807140003542984287, 0.45355417259430097810314927631, 0.46293893816338460349832548167, 0.61055722451858667367914297466, 0.67287019176154773191247876942, 0.67526653123164133660888894119, 0.72482237925929475243270641245, 0.74080983354681324338173521789, 0.75901491367071960693398626885, 0.76280245737677060794066122129, 0.817225582934635751764537729036, 0.859445927344910753202356439444, 0.880475687264510960124882687595, 0.897106061832670160802316306977, 0.905698032598951047770596246621, 0.910731563362321905198238877443, 0.910832898679790027680075763417, 0.932098302050877510774054579851, 0.956712825070201278336939414552

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.