# Properties

 Label 8-55e4-1.1-c3e4-0-1 Degree $8$ Conductor $9150625$ Sign $1$ Analytic cond. $110.895$ Root an. cond. $1.80141$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 9·3-s − 6·4-s + 20·5-s + 9·6-s + 9·7-s + 2·8-s + 11·9-s + 20·10-s + 44·11-s − 54·12-s + 70·13-s + 9·14-s + 180·15-s − 21·16-s + 103·17-s + 11·18-s − 205·19-s − 120·20-s + 81·21-s + 44·22-s − 56·23-s + 18·24-s + 250·25-s + 70·26-s − 90·27-s − 54·28-s + ⋯
 L(s)  = 1 + 0.353·2-s + 1.73·3-s − 3/4·4-s + 1.78·5-s + 0.612·6-s + 0.485·7-s + 0.0883·8-s + 0.407·9-s + 0.632·10-s + 1.20·11-s − 1.29·12-s + 1.49·13-s + 0.171·14-s + 3.09·15-s − 0.328·16-s + 1.46·17-s + 0.144·18-s − 2.47·19-s − 1.34·20-s + 0.841·21-s + 0.426·22-s − 0.507·23-s + 0.153·24-s + 2·25-s + 0.528·26-s − 0.641·27-s − 0.364·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$9150625$$    =    $$5^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$110.895$$ Root analytic conductor: $$1.80141$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 9150625,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$7.274868421$$ $$L(\frac12)$$ $$\approx$$ $$7.274868421$$ $$L(\frac{5}{2})$$ 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$C_1$ $$( 1 - p T )^{4}$$
11$C_1$ $$( 1 - p T )^{4}$$
good2$C_2 \wr S_4$ $$1 - T + 7 T^{2} - 15 T^{3} + 5 p^{4} T^{4} - 15 p^{3} T^{5} + 7 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8}$$
3$C_2 \wr S_4$ $$1 - p^{2} T + 70 T^{2} - 49 p^{2} T^{3} + 2674 T^{4} - 49 p^{5} T^{5} + 70 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8}$$
7$C_2 \wr S_4$ $$1 - 9 T + 708 T^{2} - 7885 T^{3} + 337686 T^{4} - 7885 p^{3} T^{5} + 708 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr S_4$ $$1 - 70 T + 6048 T^{2} - 213922 T^{3} + 14034062 T^{4} - 213922 p^{3} T^{5} + 6048 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr S_4$ $$1 - 103 T + 13792 T^{2} - 887937 T^{3} + 87444206 T^{4} - 887937 p^{3} T^{5} + 13792 p^{6} T^{6} - 103 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 + 205 T + 34976 T^{2} + 3873885 T^{3} + 380485006 T^{4} + 3873885 p^{3} T^{5} + 34976 p^{6} T^{6} + 205 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr S_4$ $$1 + 56 T + 2040 p T^{2} + 1999704 T^{3} + 845457934 T^{4} + 1999704 p^{3} T^{5} + 2040 p^{7} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 + 79 T + 26818 T^{2} + 3964293 T^{3} + 1224657242 T^{4} + 3964293 p^{3} T^{5} + 26818 p^{6} T^{6} + 79 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 - 49 T + 94508 T^{2} - 3937677 T^{3} + 3982227494 T^{4} - 3937677 p^{3} T^{5} + 94508 p^{6} T^{6} - 49 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - 289 T + 76410 T^{2} + 5380301 T^{3} - 1483114390 T^{4} + 5380301 p^{3} T^{5} + 76410 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 - 736 T + 441052 T^{2} - 165296160 T^{3} + 51625016486 T^{4} - 165296160 p^{3} T^{5} + 441052 p^{6} T^{6} - 736 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 + 152 T + 158892 T^{2} + 27361880 T^{3} + 15212838326 T^{4} + 27361880 p^{3} T^{5} + 158892 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 - 412 T + 138336 T^{2} - 73140204 T^{3} + 27137622526 T^{4} - 73140204 p^{3} T^{5} + 138336 p^{6} T^{6} - 412 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 1685 T + 1590202 T^{2} - 996012543 T^{3} + 449858930474 T^{4} - 996012543 p^{3} T^{5} + 1590202 p^{6} T^{6} - 1685 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 + 842 T + 352020 T^{2} - 46490886 T^{3} - 63133979162 T^{4} - 46490886 p^{3} T^{5} + 352020 p^{6} T^{6} + 842 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 + 1097 T + 1346310 T^{2} + 822011171 T^{3} + 512711118962 T^{4} + 822011171 p^{3} T^{5} + 1346310 p^{6} T^{6} + 1097 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 + 122 T - 57444 T^{2} - 8936710 T^{3} + 101266983734 T^{4} - 8936710 p^{3} T^{5} - 57444 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 + 521 T + 927280 T^{2} + 492331245 T^{3} + 408706607486 T^{4} + 492331245 p^{3} T^{5} + 927280 p^{6} T^{6} + 521 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 + 590 T + 565128 T^{2} + 6063986 p T^{3} + 275503933262 T^{4} + 6063986 p^{4} T^{5} + 565128 p^{6} T^{6} + 590 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 + 1118 T + 1188812 T^{2} + 493112646 T^{3} + 396666815654 T^{4} + 493112646 p^{3} T^{5} + 1188812 p^{6} T^{6} + 1118 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 + 122 T + 876132 T^{2} + 323762010 T^{3} + 688436592166 T^{4} + 323762010 p^{3} T^{5} + 876132 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 + 181 T + 2279242 T^{2} + 437587107 T^{3} + 2247142370474 T^{4} + 437587107 p^{3} T^{5} + 2279242 p^{6} T^{6} + 181 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 - 1474 T + 3318960 T^{2} - 3496771774 T^{3} + 4536975597470 T^{4} - 3496771774 p^{3} T^{5} + 3318960 p^{6} T^{6} - 1474 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$