# Properties

 Label 8-13e4-1.1-c3e4-0-0 Degree $8$ Conductor $28561$ Sign $1$ Analytic cond. $0.346128$ Root an. cond. $0.875799$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5·2-s − 5·3-s + 18·4-s − 30·5-s − 25·6-s − 15·7-s + 65·8-s + 22·9-s − 150·10-s − 17·11-s − 90·12-s + 125·13-s − 75·14-s + 150·15-s + 189·16-s − 70·17-s + 110·18-s + 141·19-s − 540·20-s + 75·21-s − 85·22-s − 145·23-s − 325·24-s + 275·25-s + 625·26-s + 65·27-s − 270·28-s + ⋯
 L(s)  = 1 + 1.76·2-s − 0.962·3-s + 9/4·4-s − 2.68·5-s − 1.70·6-s − 0.809·7-s + 2.87·8-s + 0.814·9-s − 4.74·10-s − 0.465·11-s − 2.16·12-s + 2.66·13-s − 1.43·14-s + 2.58·15-s + 2.95·16-s − 0.998·17-s + 1.44·18-s + 1.70·19-s − 6.03·20-s + 0.779·21-s − 0.823·22-s − 1.31·23-s − 2.76·24-s + 11/5·25-s + 4.71·26-s + 0.463·27-s − 1.82·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$28561$$    =    $$13^{4}$$ Sign: $1$ Analytic conductor: $$0.346128$$ Root analytic conductor: $$0.875799$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 28561,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.146919712$$ $$L(\frac12)$$ $$\approx$$ $$1.146919712$$ $$L(\frac{5}{2})$$ 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)$
bad13$C_2^2$ $$1 - 125 T + 612 p T^{2} - 125 p^{3} T^{3} + p^{6} T^{4}$$
good2$D_4\times C_2$ $$1 - 5 T + 7 T^{2} - 5 p T^{3} + 15 p^{2} T^{4} - 5 p^{4} T^{5} + 7 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8}$$
3$D_4\times C_2$ $$1 + 5 T + p T^{2} - 160 T^{3} - 920 T^{4} - 160 p^{3} T^{5} + p^{7} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8}$$
5$C_2^2$ $$( 1 + 3 p T + 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} )^{2}$$
7$D_4\times C_2$ $$1 + 15 T - 513 T^{2} + 780 T^{3} + 349820 T^{4} + 780 p^{3} T^{5} - 513 p^{6} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8}$$
11$D_4\times C_2$ $$1 + 17 T - 1489 T^{2} - 15028 T^{3} + 1005064 T^{4} - 15028 p^{3} T^{5} - 1489 p^{6} T^{6} + 17 p^{9} T^{7} + p^{12} T^{8}$$
17$D_4\times C_2$ $$1 + 70 T - 5063 T^{2} + 9590 T^{3} + 51050100 T^{4} + 9590 p^{3} T^{5} - 5063 p^{6} T^{6} + 70 p^{9} T^{7} + p^{12} T^{8}$$
19$D_4\times C_2$ $$1 - 141 T + 1299 T^{2} - 36096 p T^{3} + 448424 p^{2} T^{4} - 36096 p^{4} T^{5} + 1299 p^{6} T^{6} - 141 p^{9} T^{7} + p^{12} T^{8}$$
23$D_4\times C_2$ $$1 + 145 T - 3937 T^{2} + 91060 T^{3} + 219254380 T^{4} + 91060 p^{3} T^{5} - 3937 p^{6} T^{6} + 145 p^{9} T^{7} + p^{12} T^{8}$$
29$D_4\times C_2$ $$1 + 34 T - 32611 T^{2} - 510374 T^{3} + 517193284 T^{4} - 510374 p^{3} T^{5} - 32611 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8}$$
31$D_{4}$ $$( 1 + 140 T + 21982 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 190 T - 66003 T^{2} - 151430 T^{3} + 6030722900 T^{4} - 151430 p^{3} T^{5} - 66003 p^{6} T^{6} - 190 p^{9} T^{7} + p^{12} T^{8}$$
41$D_4\times C_2$ $$1 + 538 T + 80941 T^{2} + 38015618 T^{3} + 18774626844 T^{4} + 38015618 p^{3} T^{5} + 80941 p^{6} T^{6} + 538 p^{9} T^{7} + p^{12} T^{8}$$
43$D_4\times C_2$ $$1 + 455 T + 36243 T^{2} + 5354440 T^{3} + 6385191800 T^{4} + 5354440 p^{3} T^{5} + 36243 p^{6} T^{6} + 455 p^{9} T^{7} + p^{12} T^{8}$$
47$D_{4}$ $$( 1 - 60 T + 125246 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
53$D_{4}$ $$( 1 - 545 T + 256304 T^{2} - 545 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 809 T + 92959 T^{2} - 121968076 T^{3} + 138709769544 T^{4} - 121968076 p^{3} T^{5} + 92959 p^{6} T^{6} - 809 p^{9} T^{7} + p^{12} T^{8}$$
61$D_4\times C_2$ $$1 + 502 T - 94959 T^{2} - 53713498 T^{3} + 11662829084 T^{4} - 53713498 p^{3} T^{5} - 94959 p^{6} T^{6} + 502 p^{9} T^{7} + p^{12} T^{8}$$
67$D_4\times C_2$ $$1 - 475 T - 397113 T^{2} - 10075700 T^{3} + 229484582600 T^{4} - 10075700 p^{3} T^{5} - 397113 p^{6} T^{6} - 475 p^{9} T^{7} + p^{12} T^{8}$$
71$D_4\times C_2$ $$1 + 127 T - 656869 T^{2} - 5438648 T^{3} + 319053277564 T^{4} - 5438648 p^{3} T^{5} - 656869 p^{6} T^{6} + 127 p^{9} T^{7} + p^{12} T^{8}$$
73$D_{4}$ $$( 1 - 585 T + 832884 T^{2} - 585 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_{4}$ $$( 1 - 260 T + 1117974 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 921 T - 707351 T^{2} + 134147334 T^{3} + 1324901569974 T^{4} + 134147334 p^{3} T^{5} - 707351 p^{6} T^{6} + 921 p^{9} T^{7} + p^{12} T^{8}$$
97$D_4\times C_2$ $$1 - 415 T - 761343 T^{2} + 370087870 T^{3} - 118607901730 T^{4} + 370087870 p^{3} T^{5} - 761343 p^{6} T^{6} - 415 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$