# Properties

 Label 8-17e8-1.1-c3e4-0-1 Degree $8$ Conductor $6975757441$ Sign $1$ Analytic cond. $84538.7$ Root an. cond. $4.12935$ 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 − 2·3-s − 10·4-s + 14·5-s − 2·6-s + 36·7-s + 7·8-s − 49·9-s + 14·10-s + 10·11-s + 20·12-s − 22·13-s + 36·14-s − 28·15-s + 61·16-s − 49·18-s + 22·19-s − 140·20-s − 72·21-s + 10·22-s + 380·23-s − 14·24-s + 37·25-s − 22·26-s − 46·27-s − 360·28-s − 78·29-s + ⋯
 L(s)  = 1 + 0.353·2-s − 0.384·3-s − 5/4·4-s + 1.25·5-s − 0.136·6-s + 1.94·7-s + 0.309·8-s − 1.81·9-s + 0.442·10-s + 0.274·11-s + 0.481·12-s − 0.469·13-s + 0.687·14-s − 0.481·15-s + 0.953·16-s − 0.641·18-s + 0.265·19-s − 1.56·20-s − 0.748·21-s + 0.0969·22-s + 3.44·23-s − 0.119·24-s + 0.295·25-s − 0.165·26-s − 0.327·27-s − 2.42·28-s − 0.499·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$17^{8}$$ Sign: $1$ Analytic conductor: $$84538.7$$ Root analytic conductor: $$4.12935$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 17^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$8.522080152$$ $$L(\frac12)$$ $$\approx$$ $$8.522080152$$ $$L(\frac{5}{2})$$ 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)$
bad17 $$1$$
good2$C_2 \wr S_4$ $$1 - T + 11 T^{2} - 7 p^{2} T^{3} + 21 p^{2} T^{4} - 7 p^{5} T^{5} + 11 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8}$$
3$C_2 \wr S_4$ $$1 + 2 T + 53 T^{2} + 250 T^{3} + 163 p^{2} T^{4} + 250 p^{3} T^{5} + 53 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8}$$
5$C_2 \wr S_4$ $$1 - 14 T + 159 T^{2} + 1576 T^{3} - 20291 T^{4} + 1576 p^{3} T^{5} + 159 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8}$$
7$C_2 \wr S_4$ $$1 - 36 T + 831 T^{2} - 5778 T^{3} + 58279 T^{4} - 5778 p^{3} T^{5} + 831 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8}$$
11$C_2 \wr S_4$ $$1 - 10 T + 3568 T^{2} - 40200 T^{3} + 6271361 T^{4} - 40200 p^{3} T^{5} + 3568 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr S_4$ $$1 + 22 T + 5279 T^{2} + 164240 T^{3} + 13515729 T^{4} + 164240 p^{3} T^{5} + 5279 p^{6} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 - 22 T + 18245 T^{2} - 624006 T^{3} + 160564655 T^{4} - 624006 p^{3} T^{5} + 18245 p^{6} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr S_4$ $$1 - 380 T + 85875 T^{2} - 13106270 T^{3} + 1616857555 T^{4} - 13106270 p^{3} T^{5} + 85875 p^{6} T^{6} - 380 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 + 78 T + 94851 T^{2} + 5618376 T^{3} + 3436887625 T^{4} + 5618376 p^{3} T^{5} + 94851 p^{6} T^{6} + 78 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 - 362 T + 152360 T^{2} - 32276128 T^{3} + 7264494729 T^{4} - 32276128 p^{3} T^{5} + 152360 p^{6} T^{6} - 362 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - 512 T + 190856 T^{2} - 50680960 T^{3} + 12444415950 T^{4} - 50680960 p^{3} T^{5} + 190856 p^{6} T^{6} - 512 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 - 840 T + 522636 T^{2} - 4959288 p T^{3} + 63721821766 T^{4} - 4959288 p^{4} T^{5} + 522636 p^{6} T^{6} - 840 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 + 114 T + 161220 T^{2} + 11775000 T^{3} + 17345138341 T^{4} + 11775000 p^{3} T^{5} + 161220 p^{6} T^{6} + 114 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 - 10 T + 329476 T^{2} + 4191888 T^{3} + 46702622069 T^{4} + 4191888 p^{3} T^{5} + 329476 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 50 T + 223267 T^{2} + 23693064 T^{3} + 25976703257 T^{4} + 23693064 p^{3} T^{5} + 223267 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 - 996 T + 958284 T^{2} - 541470132 T^{3} + 301264321318 T^{4} - 541470132 p^{3} T^{5} + 958284 p^{6} T^{6} - 996 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 - 448 T + 528746 T^{2} - 191029344 T^{3} + 163720487563 T^{4} - 191029344 p^{3} T^{5} + 528746 p^{6} T^{6} - 448 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 + 868 T + 962944 T^{2} + 605911940 T^{3} + 428616419390 T^{4} + 605911940 p^{3} T^{5} + 962944 p^{6} T^{6} + 868 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 - 1116 T + 1798284 T^{2} - 1235308700 T^{3} + 1031640282550 T^{4} - 1235308700 p^{3} T^{5} + 1798284 p^{6} T^{6} - 1116 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 - 540 T + 1193894 T^{2} - 347390120 T^{3} + 579821039015 T^{4} - 347390120 p^{3} T^{5} + 1193894 p^{6} T^{6} - 540 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 - 940 T + 932284 T^{2} - 641029836 T^{3} + 662196409398 T^{4} - 641029836 p^{3} T^{5} + 932284 p^{6} T^{6} - 940 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 - 850 T + 1884977 T^{2} - 1471992022 T^{3} + 1503367753131 T^{4} - 1471992022 p^{3} T^{5} + 1884977 p^{6} T^{6} - 850 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 - 784 T + 2377308 T^{2} - 1342609264 T^{3} + 2382598755046 T^{4} - 1342609264 p^{3} T^{5} + 2377308 p^{6} T^{6} - 784 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 + 518 T + 274919 T^{2} - 5296380 T^{3} + 1036759053865 T^{4} - 5296380 p^{3} T^{5} + 274919 p^{6} T^{6} + 518 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$