# Properties

 Label 8-1904e4-1.1-c1e4-0-4 Degree $8$ Conductor $1.314\times 10^{13}$ Sign $1$ Analytic cond. $53428.8$ Root an. cond. $3.89916$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·3-s − 5-s + 4·7-s + 2·9-s − 4·11-s − 8·13-s + 3·15-s + 4·17-s − 14·19-s − 12·21-s − 8·23-s − 4·25-s + 2·27-s + 4·29-s − 5·31-s + 12·33-s − 4·35-s − 4·37-s + 24·39-s − 7·41-s − 19·43-s − 2·45-s − 8·47-s + 10·49-s − 12·51-s + 5·53-s + 4·55-s + ⋯
 L(s)  = 1 − 1.73·3-s − 0.447·5-s + 1.51·7-s + 2/3·9-s − 1.20·11-s − 2.21·13-s + 0.774·15-s + 0.970·17-s − 3.21·19-s − 2.61·21-s − 1.66·23-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 0.898·31-s + 2.08·33-s − 0.676·35-s − 0.657·37-s + 3.84·39-s − 1.09·41-s − 2.89·43-s − 0.298·45-s − 1.16·47-s + 10/7·49-s − 1.68·51-s + 0.686·53-s + 0.539·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 17^{4}\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 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 7^{4} \cdot 17^{4}$$ Sign: $1$ Analytic conductor: $$53428.8$$ Root analytic conductor: $$3.89916$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{16} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_1$ $$( 1 - T )^{4}$$
17$C_1$ $$( 1 - T )^{4}$$
good3$(((C_4 \times C_2): C_2):C_2):C_2$ $$1 + p T + 7 T^{2} + 13 T^{3} + 28 T^{4} + 13 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
5$C_2 \wr C_2\wr C_2$ $$1 + T + p T^{2} - 7 T^{3} + 4 T^{4} - 7 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$D_{4}$ $$( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 14 T + 116 T^{2} + 710 T^{3} + 3510 T^{4} + 710 p T^{5} + 116 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 72 T^{2} + 328 T^{3} + 1998 T^{4} + 328 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 48 T^{2} - 364 T^{3} + 1118 T^{4} - 364 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 + 5 T + 39 T^{2} - 35 T^{3} + 96 T^{4} - 35 p T^{5} + 39 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 80 T^{2} + 140 T^{3} + 2878 T^{4} + 140 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 7 T + 99 T^{2} + 625 T^{3} + 5720 T^{4} + 625 p T^{5} + 99 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 19 T + 237 T^{2} + 2183 T^{3} + 16508 T^{4} + 2183 p T^{5} + 237 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 76 T^{2} + 616 T^{3} + 5542 T^{4} + 616 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 5 T + 111 T^{2} - 575 T^{3} + 6632 T^{4} - 575 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 192 T^{2} - 80 T^{3} + 15758 T^{4} - 80 p T^{5} + 192 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 23 T + 357 T^{2} + 3847 T^{3} + 34148 T^{4} + 3847 p T^{5} + 357 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 15 T + 239 T^{2} + 2555 T^{3} + 22872 T^{4} + 2555 p T^{5} + 239 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 152 T^{2} + 1018 T^{3} + 10798 T^{4} + 1018 p T^{5} + 152 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 5 T + 91 T^{2} + 155 T^{3} + 2872 T^{4} + 155 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 24 T + 396 T^{2} - 4920 T^{3} + 49830 T^{4} - 4920 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 10 T + 212 T^{2} + 1890 T^{3} + 25814 T^{4} + 1890 p T^{5} + 212 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 16 T + 136 T^{2} + 1280 T^{3} + 16110 T^{4} + 1280 p T^{5} + 136 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 15 T + 323 T^{2} + 2665 T^{3} + 37944 T^{4} + 2665 p T^{5} + 323 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$