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Properties

Label 8-12e8-1.1-c12e4-0-2
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $3.00070\times 10^{8}$
Root an. cond. $11.4723$
Motivic weight $12$
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  + 1.49e4·5-s + 1.25e7·13-s − 5.74e7·17-s − 9.42e7·25-s + 5.97e8·29-s − 3.79e9·37-s + 1.14e10·41-s + 2.79e10·49-s − 9.45e10·53-s + 8.74e10·61-s + 1.87e11·65-s − 7.59e11·73-s − 8.55e11·85-s + 6.81e11·89-s − 4.07e11·97-s − 1.04e12·101-s − 4.91e11·109-s + 5.58e12·113-s + 8.71e12·121-s − 5.37e11·125-s + 127-s + 131-s + 137-s + 139-s + 8.90e12·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.953·5-s + 2.60·13-s − 2.37·17-s − 0.386·25-s + 1.00·29-s − 1.47·37-s + 2.41·41-s + 2.02·49-s − 4.26·53-s + 1.69·61-s + 2.48·65-s − 5.01·73-s − 2.26·85-s + 1.37·89-s − 0.489·97-s − 0.982·101-s − 0.293·109-s + 2.68·113-s + 2.77·121-s − 0.140·125-s + 0.958·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.00070\times 10^{8}\)
Root analytic conductor: \(11.4723\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 6, 6, 6, 6 ),\ 1 )\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.514351509\)
\(L(\frac12)\) \(\approx\) \(2.514351509\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_{4}$ \( ( 1 - 7452 T + 1043422 p^{3} T^{2} - 7452 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 27997958788 T^{2} + 10802140476862671462 p^{2} T^{4} - 27997958788 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 72063054820 p^{2} T^{2} + \)\(25\!\cdots\!02\)\( p^{4} T^{4} - 72063054820 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 - 6276260 T + 34493357065638 T^{2} - 6276260 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 28714500 T + 697893999497606 T^{2} + 28714500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 72628763313620 p T^{2} + \)\(76\!\cdots\!42\)\( T^{4} + 72628763313620 p^{25} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 4191099606953860 T^{2} + \)\(49\!\cdots\!82\)\( T^{4} - 4191099606953860 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 - 298725084 T + 415040300804049446 T^{2} - 298725084 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 311340683481463300 T^{2} + \)\(62\!\cdots\!42\)\( T^{4} - 311340683481463300 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 + 1896332060 T + 11650287819649513062 T^{2} + 1896332060 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 5740294716 T + 50715457869976185926 T^{2} - 5740294716 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 49567287012768743620 T^{2} + \)\(37\!\cdots\!02\)\( T^{4} - 49567287012768743620 p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(38\!\cdots\!28\)\( T^{2} + \)\(63\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!28\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 + 47278891620 T + \)\(15\!\cdots\!26\)\( T^{2} + 47278891620 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(54\!\cdots\!60\)\( T^{2} + \)\(13\!\cdots\!22\)\( T^{4} - \)\(54\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 - 43720711972 T + \)\(18\!\cdots\!38\)\( T^{2} - 43720711972 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!42\)\( T^{4} - \)\(15\!\cdots\!80\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(19\!\cdots\!28\)\( T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!28\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 + 379709948860 T + \)\(80\!\cdots\!46\)\( T^{2} + 379709948860 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(23\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!86\)\( T^{4} - \)\(23\!\cdots\!64\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(40\!\cdots\!48\)\( T^{2} + \)\(63\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 - 340918906428 T + \)\(46\!\cdots\!38\)\( T^{2} - 340918906428 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 203900645500 T + \)\(10\!\cdots\!38\)\( T^{2} + 203900645500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.45144352473171807615881292626, −6.95540043417581410953910605082, −6.70943828812978344728679791515, −6.33071944238117717343532186645, −6.26135788123638376292968846578, −6.03772214376852188478809928361, −5.76869182674882525467070988604, −5.54534517583841368994433865775, −5.09866525685664569123261239834, −4.78741717301884855828025476689, −4.29698396420731686739971041702, −4.24052882425922978492084207028, −4.13329450646427644438108277378, −3.66641362859881883115853919696, −3.11960429195725379710326550782, −2.94960905711478002724138831541, −2.92671497583450682457573514995, −2.14011224093759539016698039977, −2.05695317541485590859994157964, −1.81856932763136501949965740205, −1.49542487186043181592854455690, −1.19038640996400252565058358217, −0.804623767355836101377511704823, −0.58517421340189649851646362328, −0.14280163503293541296763472370, 0.14280163503293541296763472370, 0.58517421340189649851646362328, 0.804623767355836101377511704823, 1.19038640996400252565058358217, 1.49542487186043181592854455690, 1.81856932763136501949965740205, 2.05695317541485590859994157964, 2.14011224093759539016698039977, 2.92671497583450682457573514995, 2.94960905711478002724138831541, 3.11960429195725379710326550782, 3.66641362859881883115853919696, 4.13329450646427644438108277378, 4.24052882425922978492084207028, 4.29698396420731686739971041702, 4.78741717301884855828025476689, 5.09866525685664569123261239834, 5.54534517583841368994433865775, 5.76869182674882525467070988604, 6.03772214376852188478809928361, 6.26135788123638376292968846578, 6.33071944238117717343532186645, 6.70943828812978344728679791515, 6.95540043417581410953910605082, 7.45144352473171807615881292626

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