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Properties

Label 8-48e4-1.1-c12e4-0-4
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.70457\times 10^{6}$
Root an. cond. $6.62357$
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  + 2.23e4·5-s − 3.54e5·9-s + 8.90e6·13-s + 7.25e7·17-s + 1.21e8·25-s − 7.28e8·29-s − 2.72e9·37-s − 7.16e9·41-s − 7.93e9·45-s + 4.73e10·49-s + 3.26e10·53-s − 1.30e11·61-s + 1.99e11·65-s − 1.70e11·73-s + 9.41e10·81-s + 1.62e12·85-s + 2.01e12·89-s + 1.49e12·97-s − 1.78e11·101-s − 3.70e12·109-s + 1.95e12·113-s − 3.15e12·117-s + 2.24e11·121-s + 2.52e12·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.43·5-s − 2/3·9-s + 1.84·13-s + 3.00·17-s + 0.497·25-s − 1.22·29-s − 1.06·37-s − 1.50·41-s − 0.955·45-s + 3.42·49-s + 1.47·53-s − 2.53·61-s + 2.64·65-s − 1.12·73-s + 1/3·81-s + 4.30·85-s + 4.04·89-s + 1.79·97-s − 0.168·101-s − 2.20·109-s + 0.939·113-s − 1.23·117-s + 0.0716·121-s + 0.663·125-s − 1.75·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.70457\times 10^{6}\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(10.83216304\)
\(L(\frac12)\) \(\approx\) \(10.83216304\)
\(L(7)\) 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$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 11196 T + 5091542 p^{2} T^{2} - 11196 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 47395207588 T^{2} + 19089410985529124262 p^{2} T^{4} - 47395207588 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 1858523140 p^{2} T^{2} - \)\(10\!\cdots\!98\)\( p^{4} T^{4} - 1858523140 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 - 4452980 T + 51426312946566 T^{2} - 4452980 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 36273780 T + 1481760659979686 T^{2} - 36273780 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5850927212613988 T^{2} + \)\(16\!\cdots\!78\)\( T^{4} - 5850927212613988 p^{24} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 61390384196685700 T^{2} + \)\(18\!\cdots\!82\)\( T^{4} - 61390384196685700 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 + 364205556 T + 599383288421216966 T^{2} + 364205556 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 404370170843358244 T^{2} - \)\(33\!\cdots\!74\)\( T^{4} - 404370170843358244 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 + 1362286940 T - 1500398694640026138 T^{2} + 1362286940 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 3582060300 T + 39700473140168754662 T^{2} + 3582060300 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 10651747351856438300 T^{2} - \)\(19\!\cdots\!98\)\( T^{4} + 10651747351856438300 p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!88\)\( T^{2} + \)\(21\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!88\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 - 16309635660 T + \)\(10\!\cdots\!18\)\( T^{2} - 16309635660 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(46\!\cdots\!60\)\( T^{2} + \)\(54\!\cdots\!22\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 + 65368120892 T + \)\(39\!\cdots\!58\)\( T^{2} + 65368120892 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!48\)\( T^{2} + \)\(75\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(96\!\cdots\!40\)\( T^{2} + \)\(91\!\cdots\!62\)\( T^{4} - \)\(96\!\cdots\!40\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 + 85248724540 T + \)\(33\!\cdots\!66\)\( T^{2} + 85248724540 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!48\)\( T^{2} + \)\(18\!\cdots\!38\)\( T^{4} - \)\(21\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!42\)\( T^{4} - \)\(21\!\cdots\!80\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 - 1006111194948 T + \)\(72\!\cdots\!18\)\( T^{2} - 1006111194948 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 745709404100 T + \)\(13\!\cdots\!78\)\( T^{2} - 745709404100 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−8.970698814260760100569394289774, −8.906327520397724757437904273454, −8.445674473146461616753960878875, −7.915184181616075614625692894276, −7.86020008167806702376805681252, −7.39876845252287960698290912098, −7.01095211596714023760234340625, −6.68978217500262640326157225388, −6.12574477178421764267341164968, −5.71637643373649772855689917971, −5.70887604822326868984061852093, −5.68887943971976167995453287556, −5.16313340331467879000062783234, −4.61739380892718109216316579266, −4.15013528920581124244699973266, −3.54792473718060594669006891489, −3.37848168564239772045451899060, −3.28220124945187454684518848555, −2.66902405040903031536507275821, −2.02600285221842485351800914694, −1.89173541206246375809364366551, −1.51258999293228993881235266824, −1.01167810417196840636569899152, −0.78026510447902557426512351568, −0.40506028679500612754103631720, 0.40506028679500612754103631720, 0.78026510447902557426512351568, 1.01167810417196840636569899152, 1.51258999293228993881235266824, 1.89173541206246375809364366551, 2.02600285221842485351800914694, 2.66902405040903031536507275821, 3.28220124945187454684518848555, 3.37848168564239772045451899060, 3.54792473718060594669006891489, 4.15013528920581124244699973266, 4.61739380892718109216316579266, 5.16313340331467879000062783234, 5.68887943971976167995453287556, 5.70887604822326868984061852093, 5.71637643373649772855689917971, 6.12574477178421764267341164968, 6.68978217500262640326157225388, 7.01095211596714023760234340625, 7.39876845252287960698290912098, 7.86020008167806702376805681252, 7.915184181616075614625692894276, 8.445674473146461616753960878875, 8.906327520397724757437904273454, 8.970698814260760100569394289774

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