Properties

Label 8-10e4-1.1-c8e4-0-0
Degree $8$
Conductor $10000$
Sign $1$
Analytic cond. $275.418$
Root an. cond. $2.01836$
Motivic weight $8$
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  − 32·2-s + 54·3-s + 512·4-s + 90·5-s − 1.72e3·6-s − 1.18e3·7-s − 4.09e3·8-s + 1.45e3·9-s − 2.88e3·10-s − 1.98e4·11-s + 2.76e4·12-s + 7.37e4·13-s + 3.79e4·14-s + 4.86e3·15-s − 1.63e4·16-s + 1.98e5·17-s − 4.66e4·18-s + 4.60e4·20-s − 6.40e4·21-s + 6.35e5·22-s + 6.31e5·23-s − 2.21e5·24-s + 2.76e5·25-s − 2.35e6·26-s + 2.05e5·27-s − 6.07e5·28-s − 1.55e5·30-s + ⋯
L(s)  = 1  − 2·2-s + 2/3·3-s + 2·4-s + 0.143·5-s − 4/3·6-s − 0.493·7-s − 8-s + 2/9·9-s − 0.287·10-s − 1.35·11-s + 4/3·12-s + 2.58·13-s + 0.987·14-s + 0.0959·15-s − 1/4·16-s + 2.38·17-s − 4/9·18-s + 0.287·20-s − 0.329·21-s + 2.71·22-s + 2.25·23-s − 2/3·24-s + 0.708·25-s − 5.16·26-s + 0.387·27-s − 0.987·28-s − 0.191·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(275.418\)
Root analytic conductor: \(2.01836\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 10000,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.423050293\)
\(L(\frac12)\) \(\approx\) \(1.423050293\)
\(L(5)\) 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$C_2$ \( ( 1 + p^{4} T + p^{7} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 18 p T - 86 p^{5} T^{2} - 18 p^{9} T^{3} + p^{16} T^{4} \)
good3$D_4\times C_2$ \( 1 - 2 p^{3} T + 2 p^{6} T^{2} - 2542 p^{4} T^{3} + 2391826 p^{2} T^{4} - 2542 p^{12} T^{5} + 2 p^{22} T^{6} - 2 p^{27} T^{7} + p^{32} T^{8} \)
7$D_4\times C_2$ \( 1 + 1186 T + 703298 T^{2} - 829181226 p T^{3} - 1340970288094 p^{2} T^{4} - 829181226 p^{9} T^{5} + 703298 p^{16} T^{6} + 1186 p^{24} T^{7} + p^{32} T^{8} \)
11$D_{4}$ \( ( 1 + 9926 T + 346521906 T^{2} + 9926 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 73704 T + 2716139808 T^{2} - 96735410875992 T^{3} + 3198469885929451454 T^{4} - 96735410875992 p^{8} T^{5} + 2716139808 p^{16} T^{6} - 73704 p^{24} T^{7} + p^{32} T^{8} \)
17$D_4\times C_2$ \( 1 - 198944 T + 19789357568 T^{2} - 2366557208509152 T^{3} + \)\(25\!\cdots\!14\)\( T^{4} - 2366557208509152 p^{8} T^{5} + 19789357568 p^{16} T^{6} - 198944 p^{24} T^{7} + p^{32} T^{8} \)
19$D_4\times C_2$ \( 1 - 41599573364 T^{2} + \)\(94\!\cdots\!86\)\( T^{4} - 41599573364 p^{16} T^{6} + p^{32} T^{8} \)
23$D_4\times C_2$ \( 1 - 631334 T + 199291309778 T^{2} - 66792152639855742 T^{3} + \)\(21\!\cdots\!14\)\( T^{4} - 66792152639855742 p^{8} T^{5} + 199291309778 p^{16} T^{6} - 631334 p^{24} T^{7} + p^{32} T^{8} \)
29$D_4\times C_2$ \( 1 - 545685235844 T^{2} + \)\(47\!\cdots\!26\)\( T^{4} - 545685235844 p^{16} T^{6} + p^{32} T^{8} \)
31$D_{4}$ \( ( 1 + 1164946 T + 1361754002586 T^{2} + 1164946 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 26388 p T + 348163272 p^{2} T^{2} - 68100577781728356 p T^{3} - \)\(23\!\cdots\!26\)\( T^{4} - 68100577781728356 p^{9} T^{5} + 348163272 p^{18} T^{6} + 26388 p^{25} T^{7} + p^{32} T^{8} \)
41$D_{4}$ \( ( 1 + 2142326 T + 7494470736786 T^{2} + 2142326 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 7146534 T + 25536474106578 T^{2} - 58215198921801907422 T^{3} + \)\(12\!\cdots\!54\)\( T^{4} - 58215198921801907422 p^{8} T^{5} + 25536474106578 p^{16} T^{6} - 7146534 p^{24} T^{7} + p^{32} T^{8} \)
47$D_4\times C_2$ \( 1 + 3300186 T + 5445613817298 T^{2} + 19872093905458278978 T^{3} - \)\(24\!\cdots\!86\)\( T^{4} + 19872093905458278978 p^{8} T^{5} + 5445613817298 p^{16} T^{6} + 3300186 p^{24} T^{7} + p^{32} T^{8} \)
53$D_4\times C_2$ \( 1 - 9989164 T + 49891698709448 T^{2} + 27564720462376467228 T^{3} - \)\(42\!\cdots\!86\)\( T^{4} + 27564720462376467228 p^{8} T^{5} + 49891698709448 p^{16} T^{6} - 9989164 p^{24} T^{7} + p^{32} T^{8} \)
59$D_4\times C_2$ \( 1 - 438743341959284 T^{2} + \)\(88\!\cdots\!46\)\( T^{4} - 438743341959284 p^{16} T^{6} + p^{32} T^{8} \)
61$D_{4}$ \( ( 1 - 42688554 T + 837683799866066 T^{2} - 42688554 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 57195466 T + 1635660665478578 T^{2} + \)\(46\!\cdots\!18\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} + \)\(46\!\cdots\!18\)\( p^{8} T^{5} + 1635660665478578 p^{16} T^{6} + 57195466 p^{24} T^{7} + p^{32} T^{8} \)
71$D_{4}$ \( ( 1 - 56502014 T + 2085017229212346 T^{2} - 56502014 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 60875756 T + 1852928834285768 T^{2} + \)\(75\!\cdots\!88\)\( T^{3} + \)\(29\!\cdots\!94\)\( T^{4} + \)\(75\!\cdots\!88\)\( p^{8} T^{5} + 1852928834285768 p^{16} T^{6} + 60875756 p^{24} T^{7} + p^{32} T^{8} \)
79$D_4\times C_2$ \( 1 - 5415311511728644 T^{2} + \)\(11\!\cdots\!26\)\( T^{4} - 5415311511728644 p^{16} T^{6} + p^{32} T^{8} \)
83$D_4\times C_2$ \( 1 - 71051214 T + 2524137505436898 T^{2} - \)\(19\!\cdots\!42\)\( T^{3} + \)\(15\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!42\)\( p^{8} T^{5} + 2524137505436898 p^{16} T^{6} - 71051214 p^{24} T^{7} + p^{32} T^{8} \)
89$D_4\times C_2$ \( 1 - 15607585398315524 T^{2} + \)\(91\!\cdots\!66\)\( T^{4} - 15607585398315524 p^{16} T^{6} + p^{32} T^{8} \)
97$D_4\times C_2$ \( 1 - 139631604 T + 9748492417806408 T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} - \)\(14\!\cdots\!52\)\( p^{8} T^{5} + 9748492417806408 p^{16} T^{6} - 139631604 p^{24} T^{7} + p^{32} T^{8} \)
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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

−14.30501672796265016763968391433, −13.54302968260372092947213546653, −13.11309136458685725931638834512, −13.05163793567470218738733518796, −12.62228437719150411911918321604, −11.88678472185384620331631544692, −11.19450287195086172685431149576, −10.94366761240132015743559132895, −10.62947254230871342329311705112, −10.12109764209585831458647257869, −9.815915125807387521427744484699, −9.209487462885946197681305327800, −8.687319578953972657443850994476, −8.610619082343812331927088025774, −8.154952320465890285594692637568, −7.31654700935727973072703631691, −7.29154380092448956159123600654, −6.47798515963746562097895777788, −5.45269018885786286140952283594, −5.34262322075842697755823825540, −3.52661646363352644481874717553, −3.43396587058692779369520740118, −2.25191905571300630546010570399, −1.09122246047220759063044200525, −0.801098543784933523304306953121, 0.801098543784933523304306953121, 1.09122246047220759063044200525, 2.25191905571300630546010570399, 3.43396587058692779369520740118, 3.52661646363352644481874717553, 5.34262322075842697755823825540, 5.45269018885786286140952283594, 6.47798515963746562097895777788, 7.29154380092448956159123600654, 7.31654700935727973072703631691, 8.154952320465890285594692637568, 8.610619082343812331927088025774, 8.687319578953972657443850994476, 9.209487462885946197681305327800, 9.815915125807387521427744484699, 10.12109764209585831458647257869, 10.62947254230871342329311705112, 10.94366761240132015743559132895, 11.19450287195086172685431149576, 11.88678472185384620331631544692, 12.62228437719150411911918321604, 13.05163793567470218738733518796, 13.11309136458685725931638834512, 13.54302968260372092947213546653, 14.30501672796265016763968391433

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