# Properties

 Degree 8 Conductor $5^{4}$ Sign $1$ Motivic weight 9 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 340·4-s + 1.14e3·5-s + 4.51e4·9-s + 1.09e5·11-s + 2.90e5·16-s − 6.36e5·19-s + 3.87e5·20-s − 1.91e4·25-s − 3.53e6·29-s − 1.05e7·31-s + 1.53e7·36-s − 1.67e7·41-s + 3.73e7·44-s + 5.15e7·45-s + 5.72e7·49-s + 1.25e8·55-s − 4.60e8·59-s + 3.60e8·61-s + 2.47e8·64-s − 4.76e7·71-s − 2.16e8·76-s − 7.28e8·79-s + 3.31e8·80-s + 9.91e8·81-s − 1.58e9·89-s − 7.26e8·95-s + 4.96e9·99-s + ⋯
 L(s)  = 1 + 0.664·4-s + 0.815·5-s + 2.29·9-s + 2.26·11-s + 1.10·16-s − 1.12·19-s + 0.541·20-s − 0.00980·25-s − 0.927·29-s − 2.05·31-s + 1.52·36-s − 0.927·41-s + 1.50·44-s + 1.87·45-s + 1.41·49-s + 1.84·55-s − 4.95·59-s + 3.33·61-s + 1.84·64-s − 0.222·71-s − 0.744·76-s − 2.10·79-s + 0.903·80-s + 2.56·81-s − 2.67·89-s − 0.914·95-s + 5.19·99-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$625$$    =    $$5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$9$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)$ $L(5)$ $\approx$ $3.92542$ $L(\frac12)$ $\approx$ $3.92542$ $L(\frac{11}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 8. If $p = 5$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad5$D_{4}$ $$1 - 228 p T + 422 p^{5} T^{2} - 228 p^{10} T^{3} + p^{18} T^{4}$$
good2$C_2^2 \wr C_2$ $$1 - 85 p^{2} T^{2} - 2733 p^{6} T^{4} - 85 p^{20} T^{6} + p^{36} T^{8}$$
3$C_2^2 \wr C_2$ $$1 - 5020 p^{2} T^{2} + 12953638 p^{4} T^{4} - 5020 p^{20} T^{6} + p^{36} T^{8}$$
7$C_2^2 \wr C_2$ $$1 - 166900 p^{3} T^{2} + 1461117678198 p^{4} T^{4} - 166900 p^{21} T^{6} + p^{36} T^{8}$$
11$D_{4}$ $$( 1 - 54984 T + 5180465446 T^{2} - 54984 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
13$C_2^2 \wr C_2$ $$1 - 35613791860 T^{2} +$$$$53\!\cdots\!58$$$$T^{4} - 35613791860 p^{18} T^{6} + p^{36} T^{8}$$
17$C_2^2 \wr C_2$ $$1 - 285780369220 T^{2} +$$$$48\!\cdots\!18$$$$T^{4} - 285780369220 p^{18} T^{6} + p^{36} T^{8}$$
19$D_{4}$ $$( 1 + 16760 p T + 505756418358 T^{2} + 16760 p^{10} T^{3} + p^{18} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 - 5779790962540 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} - 5779790962540 p^{18} T^{6} + p^{36} T^{8}$$
29$D_{4}$ $$( 1 + 1765860 T + 26950935551038 T^{2} + 1765860 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 5293856 T + 59464921598526 T^{2} + 5293856 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
37$C_2^2 \wr C_2$ $$1 - 231603274936660 T^{2} +$$$$44\!\cdots\!58$$$$T^{4} - 231603274936660 p^{18} T^{6} + p^{36} T^{8}$$
41$D_{4}$ $$( 1 + 8394276 T + 221313076168966 T^{2} + 8394276 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
43$C_2^2 \wr C_2$ $$1 - 614109141147100 T^{2} +$$$$57\!\cdots\!98$$$$T^{4} - 614109141147100 p^{18} T^{6} + p^{36} T^{8}$$
47$C_2^2 \wr C_2$ $$1 - 1368976020813580 T^{2} +$$$$29\!\cdots\!78$$$$T^{4} - 1368976020813580 p^{18} T^{6} + p^{36} T^{8}$$
53$C_2^2 \wr C_2$ $$1 - 7684297973864980 T^{2} +$$$$36\!\cdots\!78$$$$T^{4} - 7684297973864980 p^{18} T^{6} + p^{36} T^{8}$$
59$D_{4}$ $$( 1 + 230414520 T + 28555631923987078 T^{2} + 230414520 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 180245284 T + 30154717014478446 T^{2} - 180245284 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
67$C_2^2 \wr C_2$ $$1 + 41160407446058180 T^{2} +$$$$18\!\cdots\!18$$$$T^{4} + 41160407446058180 p^{18} T^{6} + p^{36} T^{8}$$
71$D_{4}$ $$( 1 + 23805936 T + 85782754020107086 T^{2} + 23805936 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
73$C_2^2 \wr C_2$ $$1 - 229489314868712740 T^{2} +$$$$37\!\cdots\!22$$$$p^{2} T^{4} - 229489314868712740 p^{18} T^{6} + p^{36} T^{8}$$
79$D_{4}$ $$( 1 + 364021760 T + 220545463862625438 T^{2} + 364021760 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 - 434569632367965820 T^{2} +$$$$10\!\cdots\!18$$$$T^{4} - 434569632367965820 p^{18} T^{6} + p^{36} T^{8}$$
89$D_{4}$ $$( 1 + 791350380 T + 832192702699668118 T^{2} + 791350380 p^{9} T^{3} + p^{18} T^{4} )^{2}$$
97$C_2^2 \wr C_2$ $$1 - 2561123777205326980 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} - 2561123777205326980 p^{18} T^{6} + p^{36} T^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}