# Properties

 Degree $8$ Conductor $390625$ Sign $1$ Motivic weight $8$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4.85e4·11-s + 7.81e4·16-s − 4.17e6·31-s − 5.28e6·41-s − 5.75e7·61-s − 9.22e7·71-s − 7.87e7·81-s + 7.22e8·101-s + 6.14e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.79e9·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 + 3.31·11-s + 1.19·16-s − 4.51·31-s − 1.87·41-s − 4.15·61-s − 3.63·71-s − 1.82·81-s + 6.94·101-s + 2.86·121-s + 3.95·176-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$390625$$    =    $$5^{8}$$ Sign: $1$ Motivic weight: $$8$$ Character: induced by $\chi_{25} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 390625,\ (\ :4, 4, 4, 4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.01948$$ $$L(\frac12)$$ $$\approx$$ $$2.01948$$ $$L(5)$$ 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 $$1$$
good2$C_2^3$ $$1 - 19543 p^{2} T^{4} + p^{32} T^{8}$$
3$C_2^3$ $$1 + 8752462 p^{2} T^{4} + p^{32} T^{8}$$
7$C_2^3$ $$1 - 22527334402 p^{4} T^{4} + p^{32} T^{8}$$
11$C_2$ $$( 1 - 12132 T + p^{8} T^{2} )^{4}$$
13$C_2^3$ $$1 + 1292678222057673218 T^{4} + p^{32} T^{8}$$
17$C_2^3$ $$1 - 92930849881804020862 T^{4} + p^{32} T^{8}$$
19$C_2^2$ $$( 1 - 5615301682 T^{2} + p^{16} T^{4} )^{2}$$
23$C_2^3$ $$1 -$$$$88\!\cdots\!22$$$$T^{4} + p^{32} T^{8}$$
29$C_2^2$ $$( 1 - 556097269022 T^{2} + p^{16} T^{4} )^{2}$$
31$C_2$ $$( 1 + 1042808 T + p^{8} T^{2} )^{4}$$
37$C_2^3$ $$1 -$$$$22\!\cdots\!82$$$$T^{4} + p^{32} T^{8}$$
41$C_2$ $$( 1 + 1321128 T + p^{8} T^{2} )^{4}$$
43$C_2^3$ $$1 -$$$$21\!\cdots\!02$$$$T^{4} + p^{32} T^{8}$$
47$C_2^3$ $$1 -$$$$30\!\cdots\!38$$$$p^{2} T^{4} + p^{32} T^{8}$$
53$C_2^3$ $$1 +$$$$32\!\cdots\!58$$$$T^{4} + p^{32} T^{8}$$
59$C_2^2$ $$( 1 - 251429853077042 T^{2} + p^{16} T^{4} )^{2}$$
61$C_2$ $$( 1 + 14393968 T + p^{8} T^{2} )^{4}$$
67$C_2^3$ $$1 -$$$$27\!\cdots\!62$$$$T^{4} + p^{32} T^{8}$$
71$C_2$ $$( 1 + 23065488 T + p^{8} T^{2} )^{4}$$
73$C_2^3$ $$1 -$$$$29\!\cdots\!22$$$$T^{4} + p^{32} T^{8}$$
79$C_2^2$ $$( 1 - 3026596265750722 T^{2} + p^{16} T^{4} )^{2}$$
83$C_2^3$ $$1 +$$$$78\!\cdots\!38$$$$T^{4} + p^{32} T^{8}$$
89$C_2^2$ $$( 1 - 7190374058544062 T^{2} + p^{16} T^{4} )^{2}$$
97$C_2^3$ $$1 -$$$$11\!\cdots\!42$$$$T^{4} + p^{32} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$