# Properties

 Label 8-13e8-1.1-c3e4-0-1 Degree $8$ Conductor $815730721$ Sign $1$ Analytic cond. $9885.78$ Root an. cond. $3.15774$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s + 58·9-s + 64·16-s − 26·17-s + 156·23-s − 78·25-s − 416·27-s − 394·29-s − 312·43-s − 256·48-s − 286·49-s + 104·51-s + 372·53-s − 290·61-s − 624·69-s + 312·75-s − 304·79-s + 1.96e3·81-s + 1.57e3·87-s − 1.63e3·101-s − 6.55e3·103-s − 1.04e3·107-s − 654·113-s − 1.63e3·121-s + 127-s + 1.24e3·129-s + 131-s + ⋯
 L(s)  = 1 − 0.769·3-s + 2.14·9-s + 16-s − 0.370·17-s + 1.41·23-s − 0.623·25-s − 2.96·27-s − 2.52·29-s − 1.10·43-s − 0.769·48-s − 0.833·49-s + 0.285·51-s + 0.964·53-s − 0.608·61-s − 1.08·69-s + 0.480·75-s − 0.432·79-s + 2.68·81-s + 1.94·87-s − 1.61·101-s − 6.26·103-s − 0.943·107-s − 0.544·113-s − 1.23·121-s + 0.000698·127-s + 0.851·129-s + 0.000666·131-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$13^{8}$$ Sign: $1$ Analytic conductor: $$9885.78$$ Root analytic conductor: $$3.15774$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{169} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.2694747472$$ $$L(\frac12)$$ $$\approx$$ $$0.2694747472$$ $$L(\frac{5}{2})$$ 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)$
bad13 $$1$$
good2$C_2^2$$\times$$C_2^2$ $$( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} )( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} )$$
3$C_2^2$ $$( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
5$C_2^2$ $$( 1 + 39 T^{2} + p^{6} T^{4} )^{2}$$
7$C_2^2$$\times$$C_2^2$ $$( 1 - 397 T^{2} + p^{6} T^{4} )( 1 + 683 T^{2} + p^{6} T^{4} )$$
11$C_2^3$ $$1 + 1638 T^{2} + 911483 T^{4} + 1638 p^{6} T^{6} + p^{12} T^{8}$$
17$C_2^2$ $$( 1 + 13 T - 4744 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
19$C_2^3$ $$1 + 12818 T^{2} + 117255243 T^{4} + 12818 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2^2$ $$( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 197 T + 14420 T^{2} + 197 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 54106 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2^3$ $$1 + 49777 T^{2} - 87976680 T^{4} + 49777 p^{6} T^{6} + p^{12} T^{8}$$
41$C_2^3$ $$1 + 110617 T^{2} + 7486016448 T^{4} + 110617 p^{6} T^{6} + p^{12} T^{8}$$
43$C_2^2$ $$( 1 + 156 T - 55171 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 181402 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2$ $$( 1 - 93 T + p^{3} T^{2} )^{4}$$
59$C_2^3$ $$1 - 335738 T^{2} + 70539471003 T^{4} - 335738 p^{6} T^{6} + p^{12} T^{8}$$
61$C_2^2$ $$( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$C_2^3$ $$1 - 141518 T^{2} - 70431037845 T^{4} - 141518 p^{6} T^{6} + p^{12} T^{8}$$
71$C_2^3$ $$1 + 288106 T^{2} - 45095216685 T^{4} + 288106 p^{6} T^{6} + p^{12} T^{8}$$
73$C_2^2$ $$( 1 - 731809 T^{2} + p^{6} T^{4} )^{2}$$
79$C_2$ $$( 1 + 76 T + p^{3} T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 749190 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^3$ $$1 + 1339182 T^{2} + 1296427138163 T^{4} + 1339182 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^3$ $$1 + 1768702 T^{2} + 2295334759875 T^{4} + 1768702 p^{6} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$