# Properties

 Label 8-13e8-1.1-c2e4-0-3 Degree $8$ Conductor $815730721$ Sign $1$ Analytic cond. $449.662$ Root an. cond. $2.14590$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·3-s + 54·9-s + 23·16-s − 168·29-s − 276·48-s − 96·53-s + 120·61-s + 216·79-s − 1.21e3·81-s + 2.01e3·87-s + 456·107-s − 504·113-s + 127-s + 131-s + 137-s + 139-s + 1.24e3·144-s + 149-s + 151-s + 157-s + 1.15e3·159-s + 163-s + 167-s + 173-s + 179-s + 181-s − 1.44e3·183-s + ⋯
 L(s)  = 1 − 4·3-s + 6·9-s + 1.43·16-s − 5.79·29-s − 5.75·48-s − 1.81·53-s + 1.96·61-s + 2.73·79-s − 15·81-s + 23.1·87-s + 4.26·107-s − 4.46·113-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 69/8·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 7.24·159-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s − 7.86·183-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$13^{8}$$ Sign: $1$ Analytic conductor: $$449.662$$ Root analytic conductor: $$2.14590$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 13^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2541972445$$ $$L(\frac12)$$ $$\approx$$ $$0.2541972445$$ $$L(2)$$ 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)$
bad13 $$1$$
good2$C_2^3$ $$1 - 23 T^{4} + p^{8} T^{8}$$
3$C_2$ $$( 1 + p T + p^{2} T^{2} )^{4}$$
5$C_2^3$ $$1 + 271 T^{4} + p^{8} T^{8}$$
7$C_2^3$ $$1 - 4801 T^{4} + p^{8} T^{8}$$
11$C_2^3$ $$1 + 82 p^{2} T^{4} + p^{8} T^{8}$$
17$C_2^2$ $$( 1 - 569 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^3$ $$1 - 154366 T^{4} + p^{8} T^{8}$$
23$C_2^2$ $$( 1 - 914 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2$ $$( 1 + 42 T + p^{2} T^{2} )^{4}$$
31$C_2^3$ $$1 - 1732798 T^{4} + p^{8} T^{8}$$
37$C_2^3$ $$1 - 3679153 T^{4} + p^{8} T^{8}$$
41$C_2^3$ $$1 - 4197086 T^{4} + p^{8} T^{8}$$
43$C_2^2$ $$( 1 - 1297 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^3$ $$1 + 9662287 T^{4} + p^{8} T^{8}$$
53$C_2$ $$( 1 + 24 T + p^{2} T^{2} )^{4}$$
59$C_2^3$ $$1 - 7045406 T^{4} + p^{8} T^{8}$$
61$C_2$ $$( 1 - 30 T + p^{2} T^{2} )^{4}$$
67$C_2^3$ $$1 + 14368994 T^{4} + p^{8} T^{8}$$
71$C_2^3$ $$1 + 40245487 T^{4} + p^{8} T^{8}$$
73$C_2^3$ $$1 + 25540994 T^{4} + p^{8} T^{8}$$
79$C_2$ $$( 1 - 54 T + p^{2} T^{2} )^{4}$$
83$C_2^3$ $$1 + 40154242 T^{4} + p^{8} T^{8}$$
89$C_2^3$ $$1 + 10424482 T^{4} + p^{8} T^{8}$$
97$C_2^3$ $$1 + 162311522 T^{4} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$