# Properties

 Label 8-3920e4-1.1-c1e4-0-0 Degree $8$ Conductor $2.361\times 10^{14}$ Sign $1$ Analytic cond. $959959.$ Root an. cond. $5.59476$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 4·5-s − 9-s − 2·11-s + 10·13-s − 8·15-s + 6·17-s + 4·23-s + 10·25-s + 2·27-s − 2·29-s + 12·31-s + 4·33-s − 20·39-s + 12·41-s + 8·43-s − 4·45-s + 2·47-s − 12·51-s − 4·53-s − 8·55-s − 8·59-s + 20·61-s + 40·65-s + 8·67-s − 8·69-s − 4·71-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1.78·5-s − 1/3·9-s − 0.603·11-s + 2.77·13-s − 2.06·15-s + 1.45·17-s + 0.834·23-s + 2·25-s + 0.384·27-s − 0.371·29-s + 2.15·31-s + 0.696·33-s − 3.20·39-s + 1.87·41-s + 1.21·43-s − 0.596·45-s + 0.291·47-s − 1.68·51-s − 0.549·53-s − 1.07·55-s − 1.04·59-s + 2.56·61-s + 4.96·65-s + 0.977·67-s − 0.963·69-s − 0.474·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 5^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$959959.$$ Root analytic conductor: $$5.59476$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3920} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$10.04376613$$ $$L(\frac12)$$ $$\approx$$ $$10.04376613$$ $$L(\frac{3}{2})$$ 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)$
bad2 $$1$$
5$C_1$ $$( 1 - T )^{4}$$
7 $$1$$
good3$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 5 T^{2} + 10 T^{3} + 26 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 3 p T^{2} + 46 T^{3} + 480 T^{4} + 46 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 6 T + p T^{2} - 62 T^{3} + 434 T^{4} - 62 p T^{5} + p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 50 T^{2} - 24 T^{3} + 1186 T^{4} - 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 54 T^{2} - 324 T^{3} + 1442 T^{4} - 324 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 134 T^{2} - 828 T^{3} + 5602 T^{4} - 828 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 106 T^{2} - 40 T^{3} + 5114 T^{4} - 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 68 T^{2} + 52 T^{3} - 2014 T^{4} + 52 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 124 T^{2} - 840 T^{3} + 7718 T^{4} - 840 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 147 T^{2} - 378 T^{3} + 9344 T^{4} - 378 p T^{5} + 147 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 142 T^{2} + 4 p T^{3} + 8866 T^{4} + 4 p^{2} T^{5} + 142 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 20 T + 288 T^{2} - 2572 T^{3} + 22350 T^{4} - 2572 p T^{5} + 288 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 154 T^{2} - 1176 T^{3} + 13994 T^{4} - 1176 p T^{5} + 154 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 32 T^{2} + 12 p T^{3} + 5438 T^{4} + 12 p^{2} T^{5} + 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 122 T^{2} + 240 T^{3} - 7038 T^{4} + 240 p T^{5} + 122 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 22 T + 389 T^{2} + 4726 T^{3} + 48924 T^{4} + 4726 p T^{5} + 389 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 36 T + 750 T^{2} + 10564 T^{3} + 110978 T^{4} + 10564 p T^{5} + 750 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 40 T + 914 T^{2} - 13800 T^{3} + 152258 T^{4} - 13800 p T^{5} + 914 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 - 26 T + 545 T^{2} - 7602 T^{3} + 85890 T^{4} - 7602 p T^{5} + 545 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$