Properties

Degree 8
Conductor $ 2^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 8
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 86·3-s + 512·4-s − 870·5-s + 2.75e3·6-s + 5.72e3·7-s + 4.09e3·8-s + 3.69e3·9-s − 2.78e4·10-s − 1.47e4·11-s + 4.40e4·12-s + 4.55e4·13-s + 1.83e5·14-s − 7.48e4·15-s − 1.63e4·16-s + 3.61e3·17-s + 1.18e5·18-s − 4.45e5·20-s + 4.92e5·21-s − 4.71e5·22-s − 4.56e5·23-s + 3.52e5·24-s − 7.83e4·25-s + 1.45e6·26-s − 2.32e3·27-s + 2.93e6·28-s − 2.39e6·30-s + ⋯
L(s)  = 1  + 2·2-s + 1.06·3-s + 2·4-s − 1.39·5-s + 2.12·6-s + 2.38·7-s + 8-s + 0.563·9-s − 2.78·10-s − 1.00·11-s + 2.12·12-s + 1.59·13-s + 4.76·14-s − 1.47·15-s − 1/4·16-s + 0.0432·17-s + 1.12·18-s − 2.78·20-s + 2.53·21-s − 2.01·22-s − 1.63·23-s + 1.06·24-s − 0.200·25-s + 3.19·26-s − 0.00436·27-s + 4.76·28-s − 2.95·30-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

\( d \)  =  \(8\)
\( N \)  =  \(10000\)    =    \(2^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(8\)
character  :  induced by $\chi_{10} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 10000,\ (\ :4, 4, 4, 4),\ 1)$
$L(\frac{9}{2})$  $\approx$  $12.0986$
$L(\frac12)$  $\approx$  $12.0986$
$L(5)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( ( 1 - p^{4} T + p^{7} T^{2} )^{2} \)
5$C_2^2$ \( 1 + 174 p T + 6682 p^{3} T^{2} + 174 p^{9} T^{3} + p^{16} T^{4} \)
good3$D_4\times C_2$ \( 1 - 86 T + 3698 T^{2} + 86 p^{3} T^{3} - 59534 p^{6} T^{4} + 86 p^{11} T^{5} + 3698 p^{16} T^{6} - 86 p^{24} T^{7} + p^{32} T^{8} \)
7$D_4\times C_2$ \( 1 - 818 p T + 334562 p^{2} T^{2} - 109347786 p^{3} T^{3} + 35482183874 p^{4} T^{4} - 109347786 p^{11} T^{5} + 334562 p^{18} T^{6} - 818 p^{25} T^{7} + p^{32} T^{8} \)
11$D_{4}$ \( ( 1 + 7366 T + 143570226 T^{2} + 7366 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 45576 T + 1038585888 T^{2} - 14387574925368 T^{3} - 50735886714949186 T^{4} - 14387574925368 p^{8} T^{5} + 1038585888 p^{16} T^{6} - 45576 p^{24} T^{7} + p^{32} T^{8} \)
17$D_4\times C_2$ \( 1 - 3616 T + 6537728 T^{2} - 25111917401568 T^{3} + 96455844642935681534 T^{4} - 25111917401568 p^{8} T^{5} + 6537728 p^{16} T^{6} - 3616 p^{24} T^{7} + p^{32} T^{8} \)
19$D_4\times C_2$ \( 1 - 1682391356 p T^{2} + \)\(65\!\cdots\!86\)\( T^{4} - 1682391356 p^{17} T^{6} + p^{32} T^{8} \)
23$D_4\times C_2$ \( 1 + 456794 T + 104330379218 T^{2} + 35544578018872962 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + 35544578018872962 p^{8} T^{5} + 104330379218 p^{16} T^{6} + 456794 p^{24} T^{7} + p^{32} T^{8} \)
29$D_4\times C_2$ \( 1 - 1965649191044 T^{2} + \)\(14\!\cdots\!26\)\( T^{4} - 1965649191044 p^{16} T^{6} + p^{32} T^{8} \)
31$D_{4}$ \( ( 1 - 2336174 T + 2953504595226 T^{2} - 2336174 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 150132 p T + 11269808712 p^{2} T^{2} + 1092082235813051196 p T^{3} + \)\(91\!\cdots\!94\)\( T^{4} + 1092082235813051196 p^{9} T^{5} + 11269808712 p^{18} T^{6} + 150132 p^{25} T^{7} + p^{32} T^{8} \)
41$D_{4}$ \( ( 1 - 5369354 T + 22297350707346 T^{2} - 5369354 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4913286 T + 12070189658898 T^{2} - 40277437086392745918 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} - 40277437086392745918 p^{8} T^{5} + 12070189658898 p^{16} T^{6} - 4913286 p^{24} T^{7} + p^{32} T^{8} \)
47$D_4\times C_2$ \( 1 + 5448474 T + 14842934464338 T^{2} - 68165833176447628158 T^{3} - \)\(10\!\cdots\!06\)\( T^{4} - 68165833176447628158 p^{8} T^{5} + 14842934464338 p^{16} T^{6} + 5448474 p^{24} T^{7} + p^{32} T^{8} \)
53$D_4\times C_2$ \( 1 - 20290316 T + 205848461689928 T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!88\)\( p^{8} T^{5} + 205848461689928 p^{16} T^{6} - 20290316 p^{24} T^{7} + p^{32} T^{8} \)
59$D_4\times C_2$ \( 1 - 96598605716084 T^{2} + \)\(45\!\cdots\!46\)\( T^{4} - 96598605716084 p^{16} T^{6} + p^{32} T^{8} \)
61$D_{4}$ \( ( 1 + 21786006 T + 431918528798546 T^{2} + 21786006 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 9518486 T + 45300787866098 T^{2} - \)\(10\!\cdots\!58\)\( T^{3} - \)\(62\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!58\)\( p^{8} T^{5} + 45300787866098 p^{16} T^{6} - 9518486 p^{24} T^{7} + p^{32} T^{8} \)
71$D_{4}$ \( ( 1 - 10203454 T + 1140007205044026 T^{2} - 10203454 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 11608364 T + 67377057378248 T^{2} - \)\(79\!\cdots\!48\)\( T^{3} - \)\(12\!\cdots\!46\)\( T^{4} - \)\(79\!\cdots\!48\)\( p^{8} T^{5} + 67377057378248 p^{16} T^{6} + 11608364 p^{24} T^{7} + p^{32} T^{8} \)
79$D_4\times C_2$ \( 1 - 3387529564746244 T^{2} + \)\(67\!\cdots\!26\)\( T^{4} - 3387529564746244 p^{16} T^{6} + p^{32} T^{8} \)
83$D_4\times C_2$ \( 1 - 64264686 T + 2064974933339298 T^{2} - \)\(15\!\cdots\!58\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} - \)\(15\!\cdots\!58\)\( p^{8} T^{5} + 2064974933339298 p^{16} T^{6} - 64264686 p^{24} T^{7} + p^{32} T^{8} \)
89$D_4\times C_2$ \( 1 - 5698613213739524 T^{2} + \)\(37\!\cdots\!66\)\( T^{4} - 5698613213739524 p^{16} T^{6} + p^{32} T^{8} \)
97$D_4\times C_2$ \( 1 - 34113396 T + 581861893326408 T^{2} - \)\(26\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(26\!\cdots\!48\)\( p^{8} T^{5} + 581861893326408 p^{16} T^{6} - 34113396 p^{24} T^{7} + p^{32} T^{8} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.92637399069533097122386065109, −13.78214722041872119100607104873, −13.62620102160974074703749549077, −13.16067869683265866034452964177, −12.25550210275362855451451000886, −12.09895903757120815560541844909, −11.92433843335304081474619900785, −11.42045660839704419276426396280, −10.97466073515949477971240202201, −10.53718074511801605687081798906, −10.12407096360998248948078790266, −8.986740064601155599554704885513, −8.648420345736611399246963327099, −7.946550086625866825545912688134, −7.81725539377555679860902636651, −7.81246732371750747120041890220, −6.48503423403572714390838654199, −5.96129765775866742918006259161, −5.22986146145030304615417834263, −4.53669008119461396286520916113, −4.21444491938491545766826703369, −3.82020611421012658503321505391, −2.84166717444066003911303309946, −2.16456074157436581167961862165, −0.986585465502458767950738782641, 0.986585465502458767950738782641, 2.16456074157436581167961862165, 2.84166717444066003911303309946, 3.82020611421012658503321505391, 4.21444491938491545766826703369, 4.53669008119461396286520916113, 5.22986146145030304615417834263, 5.96129765775866742918006259161, 6.48503423403572714390838654199, 7.81246732371750747120041890220, 7.81725539377555679860902636651, 7.946550086625866825545912688134, 8.648420345736611399246963327099, 8.986740064601155599554704885513, 10.12407096360998248948078790266, 10.53718074511801605687081798906, 10.97466073515949477971240202201, 11.42045660839704419276426396280, 11.92433843335304081474619900785, 12.09895903757120815560541844909, 12.25550210275362855451451000886, 13.16067869683265866034452964177, 13.62620102160974074703749549077, 13.78214722041872119100607104873, 13.92637399069533097122386065109

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