Properties

Label 8-48e4-1.1-c12e4-0-1
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  − 780·3-s − 1.53e5·7-s − 4.61e5·9-s + 7.25e6·13-s + 1.20e8·19-s + 1.19e8·21-s + 7.06e8·25-s + 7.79e8·27-s − 2.73e9·31-s − 1.52e7·37-s − 5.65e9·39-s − 1.62e9·43-s + 2.05e10·49-s − 9.38e10·57-s + 4.54e10·61-s + 7.06e10·63-s + 2.13e11·67-s − 2.54e11·73-s − 5.50e11·75-s − 3.08e11·79-s − 7.37e10·81-s − 1.11e12·91-s + 2.13e12·93-s + 1.27e12·97-s − 3.17e12·103-s − 2.65e12·109-s + 1.19e10·111-s + ⋯
L(s)  = 1  − 1.06·3-s − 1.30·7-s − 0.867·9-s + 1.50·13-s + 2.55·19-s + 1.39·21-s + 2.89·25-s + 2.01·27-s − 3.07·31-s − 0.00595·37-s − 1.60·39-s − 0.257·43-s + 1.48·49-s − 2.73·57-s + 0.882·61-s + 1.12·63-s + 2.35·67-s − 1.68·73-s − 3.09·75-s − 1.26·79-s − 0.261·81-s − 1.95·91-s + 3.29·93-s + 1.53·97-s − 2.65·103-s − 1.58·109-s + 0.00637·111-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

\(L(\frac{13}{2})\) \(\approx\) \(2.598299663\)
\(L(\frac12)\) \(\approx\) \(2.598299663\)
\(L(7)\) 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$C_2^2$ \( 1 + 260 p T + 4402 p^{5} T^{2} + 260 p^{13} T^{3} + p^{24} T^{4} \)
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

−8.902931288057755235869310907391, −8.705504679479900669374835524169, −8.594686656106296895468566798565, −8.121237825739202368229278584974, −7.50092557828238514687203929518, −7.33046657266286921178289451663, −6.81467072312053878905616427426, −6.69175616975046821112608257712, −6.50737372716633396021431649384, −5.73113401250693726275029910343, −5.57282861176978265143851643755, −5.44585444697943027090302901405, −5.38629986661031050281723257688, −4.51036058592273005127479312076, −4.35943841897154869606631892106, −3.49369257121475644763023352106, −3.45240446476403225288259210250, −3.04568749088862436764021877838, −3.00772925559086103748425235967, −2.28734584674266101908149833984, −1.73853302304682090355604988758, −1.25235002419660635326966873974, −0.821628816470093795766530142745, −0.63887785020506283518316494914, −0.32370843151437891384661025291, 0.32370843151437891384661025291, 0.63887785020506283518316494914, 0.821628816470093795766530142745, 1.25235002419660635326966873974, 1.73853302304682090355604988758, 2.28734584674266101908149833984, 3.00772925559086103748425235967, 3.04568749088862436764021877838, 3.45240446476403225288259210250, 3.49369257121475644763023352106, 4.35943841897154869606631892106, 4.51036058592273005127479312076, 5.38629986661031050281723257688, 5.44585444697943027090302901405, 5.57282861176978265143851643755, 5.73113401250693726275029910343, 6.50737372716633396021431649384, 6.69175616975046821112608257712, 6.81467072312053878905616427426, 7.33046657266286921178289451663, 7.50092557828238514687203929518, 8.121237825739202368229278584974, 8.594686656106296895468566798565, 8.705504679479900669374835524169, 8.902931288057755235869310907391

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