# Properties

 Label 8-75e4-1.1-c2e4-0-4 Degree $8$ Conductor $31640625$ Sign $1$ Analytic cond. $17.4415$ Root an. cond. $1.42954$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 8·4-s − 4·7-s + 12·8-s + 16·11-s + 32·13-s − 16·14-s + 15·16-s + 40·17-s + 64·22-s − 56·23-s + 128·26-s − 32·28-s − 16·31-s + 40·32-s + 160·34-s − 64·37-s − 56·41-s + 8·43-s + 128·44-s − 224·46-s − 128·47-s + 8·49-s + 256·52-s − 56·53-s − 48·56-s + 200·61-s + ⋯
 L(s)  = 1 + 2·2-s + 2·4-s − 4/7·7-s + 3/2·8-s + 1.45·11-s + 2.46·13-s − 8/7·14-s + 0.937·16-s + 2.35·17-s + 2.90·22-s − 2.43·23-s + 4.92·26-s − 8/7·28-s − 0.516·31-s + 5/4·32-s + 4.70·34-s − 1.72·37-s − 1.36·41-s + 8/43·43-s + 2.90·44-s − 4.86·46-s − 2.72·47-s + 8/49·49-s + 4.92·52-s − 1.05·53-s − 6/7·56-s + 3.27·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$31640625$$    =    $$3^{4} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$17.4415$$ Root analytic conductor: $$1.42954$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{75} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$6.106015812$$ $$L(\frac12)$$ $$\approx$$ $$6.106015812$$ $$L(2)$$ 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)$
bad3$C_2^2$ $$1 + p^{2} T^{4}$$
5 $$1$$
good2$D_4\times C_2$ $$1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{4} T^{5} + p^{7} T^{6} - p^{8} T^{7} + p^{8} T^{8}$$
7$D_4\times C_2$ $$1 + 4 T + 8 T^{2} + 156 T^{3} + 2942 T^{4} + 156 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8}$$
11$D_{4}$ $$( 1 - 8 T + 204 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 32 T + 512 T^{2} - 9120 T^{3} + 148994 T^{4} - 9120 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8}$$
17$D_4\times C_2$ $$1 - 40 T + 800 T^{2} - 15240 T^{3} + 281858 T^{4} - 15240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8}$$
19$D_4\times C_2$ $$1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8}$$
23$D_4\times C_2$ $$1 + 56 T + 1568 T^{2} + 50904 T^{3} + 1508162 T^{4} + 50904 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8}$$
29$D_4\times C_2$ $$1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8}$$
31$D_{4}$ $$( 1 + 8 T + 1722 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 + 64 T + 2048 T^{2} + 58176 T^{3} + 1440962 T^{4} + 58176 p^{2} T^{5} + 2048 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8}$$
41$D_{4}$ $$( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8}$$
47$D_4\times C_2$ $$1 + 128 T + 8192 T^{2} + 506496 T^{3} + 28260194 T^{4} + 506496 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 + 56 T + 1568 T^{2} + 155064 T^{3} + 15333122 T^{4} + 155064 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8}$$
71$C_2$ $$( 1 + 68 T + p^{2} T^{2} )^{4}$$
73$D_4\times C_2$ $$1 + 76 T + 2888 T^{2} - 65436 T^{3} - 36833458 T^{4} - 65436 p^{2} T^{5} + 2888 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8}$$
79$C_2^2$ $$( 1 - 11882 T^{2} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 16 T + 128 T^{2} - 101328 T^{3} + 79904642 T^{4} - 101328 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8}$$
97$D_4\times C_2$ $$1 - 20 T + 200 T^{2} - 173820 T^{3} + 150551438 T^{4} - 173820 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$