# Properties

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

# Origins of factors

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

## 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 $p \notin \{2,\;5\}$, $$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
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$D_4\times C_2$ $$1 - 34113396 T + 581861893326408 T^{2} -$$$$26\!\cdots\!48$$$$T^{3} +$$$$12\!\cdots\!34$$$$T^{4} -$$$$26\!\cdots\!48$$$$p^{8} T^{5} + 581861893326408 p^{16} T^{6} - 34113396 p^{24} T^{7} + p^{32} T^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}