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Properties

Label 8-5e8-1.1-c11e4-0-0
Degree $8$
Conductor $390625$
Sign $1$
Analytic cond. $136138.$
Root an. cond. $4.38275$
Motivic weight $11$
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  − 2.88e3·4-s + 3.75e5·9-s − 1.23e6·11-s + 6.59e3·16-s − 1.06e7·19-s − 1.88e8·29-s + 4.89e8·31-s − 1.08e9·36-s − 1.49e9·41-s + 3.56e9·44-s + 5.89e9·49-s − 1.46e10·59-s − 3.03e9·61-s + 1.17e10·64-s + 6.58e10·71-s + 3.06e10·76-s + 6.60e9·79-s + 5.02e10·81-s + 2.53e10·89-s − 4.63e11·99-s + 1.72e11·101-s + 3.24e10·109-s + 5.42e11·116-s + 6.56e11·121-s − 1.40e12·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.40·4-s + 2.11·9-s − 2.31·11-s + 0.00157·16-s − 0.986·19-s − 1.70·29-s + 3.06·31-s − 2.97·36-s − 2.01·41-s + 3.25·44-s + 2.98·49-s − 2.66·59-s − 0.459·61-s + 1.37·64-s + 4.33·71-s + 1.38·76-s + 0.241·79-s + 1.60·81-s + 0.481·89-s − 4.90·99-s + 1.63·101-s + 0.201·109-s + 2.39·116-s + 2.29·121-s − 4.31·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(390625\)    =    \(5^{8}\)
Sign: $1$
Analytic conductor: \(136138.\)
Root analytic conductor: \(4.38275\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 390625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.2499722272\)
\(L(\frac12)\) \(\approx\) \(0.2499722272\)
\(L(\frac{13}{2})\) 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)$
bad5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 45 p^{6} T^{2} + 129497 p^{6} T^{4} + 45 p^{28} T^{6} + p^{44} T^{8} \)
3$D_4\times C_2$ \( 1 - 375140 T^{2} + 1116800278 p^{4} T^{4} - 375140 p^{22} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 - 5896330900 T^{2} + 325445393122406502 p^{2} T^{4} - 5896330900 p^{22} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 + 618176 T + 245194892966 T^{2} + 618176 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 1140153047180 T^{2} + \)\(55\!\cdots\!38\)\( T^{4} - 1140153047180 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 - 116879825889660 T^{2} + \)\(57\!\cdots\!78\)\( T^{4} - 116879825889660 p^{22} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 + 280280 p T + 194538827137638 T^{2} + 280280 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 2072692881139220 T^{2} + \)\(28\!\cdots\!58\)\( T^{4} - 2072692881139220 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 + 3246220 p T + 23426350431097358 T^{2} + 3246220 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 244543464 T + 34393316207729486 T^{2} - 244543464 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 273812295186452780 T^{2} + \)\(81\!\cdots\!38\)\( T^{4} - 273812295186452780 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 + 745743316 T + 929792912462405846 T^{2} + 745743316 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3285052879347844100 T^{2} + \)\(43\!\cdots\!98\)\( T^{4} - 3285052879347844100 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 - 8397835342270441140 T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - 8397835342270441140 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 + 5703220206102687060 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + 5703220206102687060 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 + 7317515560 T + 55027608950440780118 T^{2} + 7317515560 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 1516425676 T + 85869525433683691566 T^{2} + 1516425676 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(60\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 - 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} - 32938471544 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!58\)\( T^{4} - \)\(66\!\cdots\!20\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 - 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} - 3302823120 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(64\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 - 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} - 12674770860 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{22} T^{6} + p^{44} T^{8} \)
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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

−10.49706045539700625338272161973, −10.40931453579657089820890694880, −9.804534365877041264034252578374, −9.440086454279273404274426650938, −9.420872857300304130147667693670, −8.622994223285981567091051344814, −8.466391440324817832190418560208, −8.071059900680152604535603076966, −7.65470269099005812462967410656, −7.31598073654435445437293975522, −6.98881822907144600133151246556, −6.23618669243862346613173439280, −6.23556168618225642900918474668, −5.17262354069329239341628840037, −5.10974393069235904714316961646, −4.75178455999600872935107285253, −4.36719791257356815192279702363, −3.91683666836421794770210924078, −3.57568837203558055188959891321, −2.70442836325544337330103686043, −2.33693670759210688802461810242, −1.90911152358226782027019613891, −1.17595278646581046360932910160, −0.71684065889097496937964072986, −0.11097168268572369597382781148, 0.11097168268572369597382781148, 0.71684065889097496937964072986, 1.17595278646581046360932910160, 1.90911152358226782027019613891, 2.33693670759210688802461810242, 2.70442836325544337330103686043, 3.57568837203558055188959891321, 3.91683666836421794770210924078, 4.36719791257356815192279702363, 4.75178455999600872935107285253, 5.10974393069235904714316961646, 5.17262354069329239341628840037, 6.23556168618225642900918474668, 6.23618669243862346613173439280, 6.98881822907144600133151246556, 7.31598073654435445437293975522, 7.65470269099005812462967410656, 8.071059900680152604535603076966, 8.466391440324817832190418560208, 8.622994223285981567091051344814, 9.420872857300304130147667693670, 9.440086454279273404274426650938, 9.804534365877041264034252578374, 10.40931453579657089820890694880, 10.49706045539700625338272161973

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