# Properties

 Label 8-245e4-1.1-c3e4-0-5 Degree $8$ Conductor $3603000625$ Sign $1$ Analytic cond. $43664.5$ Root an. cond. $3.80203$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 10·3-s + 6·4-s + 10·5-s − 20·6-s + 4·8-s + 79·9-s − 20·10-s − 66·11-s + 60·12-s − 20·13-s + 100·15-s + 24·16-s + 70·17-s − 158·18-s + 140·19-s + 60·20-s + 132·22-s + 16·23-s + 40·24-s + 25·25-s + 40·26-s + 830·27-s − 516·29-s − 200·30-s − 20·31-s + 120·32-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.92·3-s + 3/4·4-s + 0.894·5-s − 1.36·6-s + 0.176·8-s + 2.92·9-s − 0.632·10-s − 1.80·11-s + 1.44·12-s − 0.426·13-s + 1.72·15-s + 3/8·16-s + 0.998·17-s − 2.06·18-s + 1.69·19-s + 0.670·20-s + 1.27·22-s + 0.145·23-s + 0.340·24-s + 1/5·25-s + 0.301·26-s + 5.91·27-s − 3.30·29-s − 1.21·30-s − 0.115·31-s + 0.662·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$43664.5$$ Root analytic conductor: $$3.80203$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$5.282136060$$ $$L(\frac12)$$ $$\approx$$ $$5.282136060$$ $$L(\frac{5}{2})$$ 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)$
bad5$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
7 $$1$$
good2$D_4\times C_2$ $$1 + p T - p T^{2} - 5 p^{2} T^{3} - 15 p^{2} T^{4} - 5 p^{5} T^{5} - p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8}$$
3$C_2^2$ $$( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 + 6 p T + 71 p T^{2} + 498 p^{2} T^{3} + 43068 p^{2} T^{4} + 498 p^{5} T^{5} + 71 p^{7} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8}$$
13$D_{4}$ $$( 1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 70 T - 103 p T^{2} + 222250 T^{3} - 3975468 T^{4} + 222250 p^{3} T^{5} - 103 p^{7} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8}$$
19$D_4\times C_2$ $$1 - 140 T + 5382 T^{2} - 70000 T^{3} + 20669243 T^{4} - 70000 p^{3} T^{5} + 5382 p^{6} T^{6} - 140 p^{9} T^{7} + p^{12} T^{8}$$
23$D_4\times C_2$ $$1 - 16 T - 15518 T^{2} + 136960 T^{3} + 97668435 T^{4} + 136960 p^{3} T^{5} - 15518 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 258 T + 40075 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 20 T - 19682 T^{2} - 790000 T^{3} - 496133357 T^{4} - 790000 p^{3} T^{5} - 19682 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8}$$
37$D_4\times C_2$ $$1 + 328 T + 678 T^{2} + 1836800 T^{3} + 3413714075 T^{4} + 1836800 p^{3} T^{5} + 678 p^{6} T^{6} + 328 p^{9} T^{7} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 + 300 T + 50342 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_{4}$ $$( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 30 T - 189371 T^{2} - 521250 T^{3} + 25330397412 T^{4} - 521250 p^{3} T^{5} - 189371 p^{6} T^{6} + 30 p^{9} T^{7} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 + 540 T + 79346 T^{2} - 46170000 T^{3} - 20525133813 T^{4} - 46170000 p^{3} T^{5} + 79346 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8}$$
59$D_4\times C_2$ $$1 - 380 T - 284858 T^{2} - 7030000 T^{3} + 112425169323 T^{4} - 7030000 p^{3} T^{5} - 284858 p^{6} T^{6} - 380 p^{9} T^{7} + p^{12} T^{8}$$
61$D_4\times C_2$ $$1 - 1080 T + 460438 T^{2} - 272160000 T^{3} + 182111332683 T^{4} - 272160000 p^{3} T^{5} + 460438 p^{6} T^{6} - 1080 p^{9} T^{7} + p^{12} T^{8}$$
67$D_4\times C_2$ $$1 + 468 T - 335882 T^{2} - 21818160 T^{3} + 151587971355 T^{4} - 21818160 p^{3} T^{5} - 335882 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 1056 T + 949550 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 860 T + 216666 T^{2} - 219386000 T^{3} - 165591231133 T^{4} - 219386000 p^{3} T^{5} + 216666 p^{6} T^{6} + 860 p^{9} T^{7} + p^{12} T^{8}$$
79$D_4\times C_2$ $$1 + 2 p T + 364789 T^{2} - 2651806 p T^{3} - 139914650492 T^{4} - 2651806 p^{4} T^{5} + 364789 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8}$$
83$D_{4}$ $$( 1 + 40 T + 703974 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 240 T + 221662 T^{2} - 377760000 T^{3} - 510671165517 T^{4} - 377760000 p^{3} T^{5} + 221662 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8}$$
97$D_{4}$ $$( 1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$