Properties

Label 8-48e4-1.1-c12e4-0-3
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  − 300·3-s − 1.58e4·7-s + 9.34e4·9-s + 3.43e6·13-s + 2.05e6·19-s + 4.74e6·21-s + 2.69e8·25-s − 1.88e8·27-s + 2.51e9·31-s − 7.46e9·37-s − 1.02e9·39-s + 2.61e10·43-s − 5.36e10·49-s − 6.15e8·57-s − 9.30e9·61-s − 1.47e9·63-s − 8.90e10·67-s + 4.64e11·73-s − 8.08e10·75-s − 5.67e11·79-s − 1.77e11·81-s − 5.42e10·91-s − 7.55e11·93-s + 8.62e11·97-s + 1.20e12·103-s − 2.60e12·109-s + 2.24e12·111-s + ⋯
L(s)  = 1  − 0.411·3-s − 0.134·7-s + 0.175·9-s + 0.711·13-s + 0.0435·19-s + 0.0552·21-s + 1.10·25-s − 0.486·27-s + 2.83·31-s − 2.91·37-s − 0.292·39-s + 4.13·43-s − 3.87·49-s − 0.0179·57-s − 0.180·61-s − 0.0236·63-s − 0.984·67-s + 3.06·73-s − 0.454·75-s − 2.33·79-s − 0.629·81-s − 0.0954·91-s − 1.16·93-s + 1.03·97-s + 1.01·103-s − 1.55·109-s + 1.19·111-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

\(L(\frac{13}{2})\) \(\approx\) \(5.510589486\)
\(L(\frac12)\) \(\approx\) \(5.510589486\)
\(L(7)\) 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$D_{4}$ \( 1 + 100 p T - 14 p^{5} T^{2} + 100 p^{13} T^{3} + p^{24} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 53911124 p T^{2} + 143718840822246 p^{4} T^{4} - 53911124 p^{25} T^{6} + p^{48} T^{8} \)
7$D_{4}$ \( ( 1 + 7900 T + 3843493338 p T^{2} + 7900 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 327394827884 p T^{2} + \)\(11\!\cdots\!46\)\( p^{3} T^{4} - 327394827884 p^{25} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 - 1716100 T + 3125193986406 T^{2} - 1716100 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 179894628000004 T^{2} - \)\(46\!\cdots\!54\)\( T^{4} - 179894628000004 p^{24} T^{6} + p^{48} T^{8} \)
19$D_{4}$ \( ( 1 - 53948 p T + 3331057943642358 T^{2} - 53948 p^{13} T^{3} + p^{24} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 45833263225240324 T^{2} + \)\(12\!\cdots\!26\)\( T^{4} - 45833263225240324 p^{24} T^{6} + p^{48} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 652133988467730724 T^{2} + \)\(25\!\cdots\!06\)\( T^{4} - 652133988467730724 p^{24} T^{6} + p^{48} T^{8} \)
31$D_{4}$ \( ( 1 - 1259504132 T + 1639163986751557878 T^{2} - 1259504132 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 3733355900 T + 7854860827290943398 T^{2} + 3733355900 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 20975752086856231684 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - 20975752086856231684 p^{24} T^{6} + p^{48} T^{8} \)
43$D_{4}$ \( ( 1 - 13059965300 T + \)\(12\!\cdots\!86\)\( T^{2} - 13059965300 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(29\!\cdots\!04\)\( T^{2} + \)\(45\!\cdots\!66\)\( T^{4} - \)\(29\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(13\!\cdots\!04\)\( T^{2} + \)\(89\!\cdots\!66\)\( T^{4} - \)\(13\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(44\!\cdots\!64\)\( T^{2} + \)\(97\!\cdots\!46\)\( T^{4} - \)\(44\!\cdots\!64\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 + 4653617852 T + \)\(29\!\cdots\!18\)\( T^{2} + 4653617852 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 44527792300 T + \)\(16\!\cdots\!86\)\( T^{2} + 44527792300 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 + \)\(14\!\cdots\!96\)\( T^{2} + \)\(17\!\cdots\!66\)\( T^{4} + \)\(14\!\cdots\!96\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 - 232071237700 T + \)\(44\!\cdots\!78\)\( T^{2} - 232071237700 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 283511013244 T + \)\(10\!\cdots\!66\)\( T^{2} + 283511013244 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(86\!\cdots\!04\)\( T^{2} + \)\(10\!\cdots\!46\)\( T^{4} - \)\(86\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(30\!\cdots\!44\)\( T^{2} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(30\!\cdots\!44\)\( p^{24} T^{6} + p^{48} T^{8} \)
97$D_{4}$ \( ( 1 - 431479262500 T + \)\(14\!\cdots\!06\)\( T^{2} - 431479262500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−8.983791854760127952955278220814, −8.726750491806532863798388282522, −8.359549683156933934758577715274, −8.063130305240283859193427041712, −7.73888531989805003660794583344, −7.38285847873404560347430529967, −6.92510922435535049576430673573, −6.61955648931601434033083746399, −6.42738316366795149689091955584, −6.07582405843986021826280185439, −5.64378025292469873144710820413, −5.32844297997447631091314304053, −4.97376784916328939470148139548, −4.49248424043352922792303746202, −4.26951689546699708629594880087, −3.98780048864468071739503091134, −3.17840489616138815479489636266, −3.06282025190465444793469390177, −2.94051396956640476046244315843, −2.12211715443788773456193289942, −1.76241915453433969668043525945, −1.52704178392593821537966344923, −0.825986681738038834915303966092, −0.67834108436230164139687359244, −0.39375323377761732490860580526, 0.39375323377761732490860580526, 0.67834108436230164139687359244, 0.825986681738038834915303966092, 1.52704178392593821537966344923, 1.76241915453433969668043525945, 2.12211715443788773456193289942, 2.94051396956640476046244315843, 3.06282025190465444793469390177, 3.17840489616138815479489636266, 3.98780048864468071739503091134, 4.26951689546699708629594880087, 4.49248424043352922792303746202, 4.97376784916328939470148139548, 5.32844297997447631091314304053, 5.64378025292469873144710820413, 6.07582405843986021826280185439, 6.42738316366795149689091955584, 6.61955648931601434033083746399, 6.92510922435535049576430673573, 7.38285847873404560347430529967, 7.73888531989805003660794583344, 8.063130305240283859193427041712, 8.359549683156933934758577715274, 8.726750491806532863798388282522, 8.983791854760127952955278220814

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