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Properties

Label 8-6e4-1.1-c12e4-0-0
Degree $8$
Conductor $1296$
Sign $1$
Analytic cond. $904.436$
Root an. cond. $2.34178$
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  + 780·3-s − 4.09e3·4-s + 1.53e5·7-s − 4.61e5·9-s − 3.19e6·12-s + 7.25e6·13-s + 1.25e7·16-s − 1.20e8·19-s + 1.19e8·21-s + 7.06e8·25-s − 7.79e8·27-s − 6.27e8·28-s + 2.73e9·31-s + 1.88e9·36-s − 1.52e7·37-s + 5.65e9·39-s + 1.62e9·43-s + 9.81e9·48-s + 2.05e10·49-s − 2.97e10·52-s − 9.38e10·57-s + 4.54e10·61-s − 7.06e10·63-s − 3.43e10·64-s − 2.13e11·67-s − 2.54e11·73-s + 5.50e11·75-s + ⋯
L(s)  = 1  + 1.06·3-s − 4-s + 1.30·7-s − 0.867·9-s − 1.06·12-s + 1.50·13-s + 3/4·16-s − 2.55·19-s + 1.39·21-s + 2.89·25-s − 2.01·27-s − 1.30·28-s + 3.07·31-s + 0.867·36-s − 0.00595·37-s + 1.60·39-s + 0.257·43-s + 0.802·48-s + 1.48·49-s − 1.50·52-s − 2.73·57-s + 0.882·61-s − 1.12·63-s − 1/2·64-s − 2.35·67-s − 1.68·73-s + 3.09·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(904.436\)
Root analytic conductor: \(2.34178\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1296,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(3.534323749\)
\(L(\frac12)\) \(\approx\) \(3.534323749\)
\(L(7)\) 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$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
3$C_2^2$ \( 1 - 260 p T + 4402 p^{5} T^{2} - 260 p^{13} T^{3} + p^{24} T^{4} \)
good5$D_4\times C_2$ \( 1 - 706084132 T^{2} + 362268674569446 p^{4} T^{4} - 706084132 p^{24} T^{6} + p^{48} T^{8} \)
7$D_{4}$ \( ( 1 - 76540 T - 209134182 p T^{2} - 76540 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 78141673540 p^{2} T^{2} + \)\(27\!\cdots\!02\)\( p^{4} T^{4} - 78141673540 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 - 3626500 T + 23403849807462 T^{2} - 3626500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 778729264609540 T^{2} + \)\(26\!\cdots\!42\)\( T^{4} - 778729264609540 p^{24} T^{6} + p^{48} T^{8} \)
19$D_{4}$ \( ( 1 + 60134036 T + 4418229943992246 T^{2} + 60134036 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 28273646588702980 T^{2} + \)\(89\!\cdots\!82\)\( T^{4} - 28273646588702980 p^{24} T^{6} + p^{48} T^{8} \)
29$D_4\times C_2$ \( 1 - 21556270284182516 p T^{2} + \)\(21\!\cdots\!86\)\( T^{4} - 21556270284182516 p^{25} T^{6} + p^{48} T^{8} \)
31$D_{4}$ \( ( 1 - 1365863836 T + 1844733739144008246 T^{2} - 1365863836 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 7640060 T + 13154488634886924966 T^{2} + 7640060 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 79504230869787730180 T^{2} + \)\(25\!\cdots\!22\)\( T^{4} - 79504230869787730180 p^{24} T^{6} + p^{48} T^{8} \)
43$D_{4}$ \( ( 1 - 814559980 T + 47795565439841642358 T^{2} - 814559980 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - \)\(32\!\cdots\!64\)\( T^{2} + \)\(48\!\cdots\!86\)\( T^{4} - \)\(32\!\cdots\!64\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(58\!\cdots\!62\)\( T^{4} - \)\(66\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} \)
59$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(95\!\cdots\!22\)\( T^{4} - \)\(11\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 - 22738532548 T + \)\(47\!\cdots\!18\)\( T^{2} - 22738532548 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 106716804980 T + \)\(19\!\cdots\!86\)\( T^{2} + 106716804980 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + \)\(56\!\cdots\!36\)\( T^{2} - \)\(79\!\cdots\!14\)\( T^{4} + \)\(56\!\cdots\!36\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 + 127191812540 T + \)\(45\!\cdots\!42\)\( T^{2} + 127191812540 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 154290079516 T + \)\(11\!\cdots\!46\)\( T^{2} - 154290079516 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(31\!\cdots\!68\)\( T^{2} + \)\(46\!\cdots\!98\)\( T^{4} - \)\(31\!\cdots\!68\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_4\times C_2$ \( 1 - \)\(98\!\cdots\!20\)\( T^{2} + \)\(36\!\cdots\!82\)\( T^{4} - \)\(98\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} \)
97$D_{4}$ \( ( 1 - 638114237860 T + \)\(12\!\cdots\!58\)\( T^{2} - 638114237860 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−14.95709231531537142669963541456, −14.17193156421105519318947966466, −13.84819788717521938121398068858, −13.69761656897018926542940204989, −12.98220827451204007180900712829, −12.75180564268946181586688757785, −12.01373552596451827381966293520, −11.42638366898050964808187764095, −11.05463723265938482434783667408, −10.37058294287042598683520438129, −10.27260477188170514515708973493, −8.790394794587510301183306555561, −8.787728935165934326639790483133, −8.774278129596321957654829837643, −8.179749050940782613802923553780, −7.61431397023134275482323105549, −6.37300899312216697516961753252, −6.16691504109089348090783432642, −5.02441812133786742099649360934, −4.57867095536885444855804477088, −3.94161750708168357409220545920, −2.98783260521782573627115847554, −2.43520548979333277248717840343, −1.33983742148211854761494666971, −0.60671937747199961067823339247, 0.60671937747199961067823339247, 1.33983742148211854761494666971, 2.43520548979333277248717840343, 2.98783260521782573627115847554, 3.94161750708168357409220545920, 4.57867095536885444855804477088, 5.02441812133786742099649360934, 6.16691504109089348090783432642, 6.37300899312216697516961753252, 7.61431397023134275482323105549, 8.179749050940782613802923553780, 8.774278129596321957654829837643, 8.787728935165934326639790483133, 8.790394794587510301183306555561, 10.27260477188170514515708973493, 10.37058294287042598683520438129, 11.05463723265938482434783667408, 11.42638366898050964808187764095, 12.01373552596451827381966293520, 12.75180564268946181586688757785, 12.98220827451204007180900712829, 13.69761656897018926542940204989, 13.84819788717521938121398068858, 14.17193156421105519318947966466, 14.95709231531537142669963541456

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