# Properties

 Label 8-338e4-1.1-c3e4-0-7 Degree $8$ Conductor $13051691536$ Sign $1$ Analytic cond. $158172.$ Root an. cond. $4.46571$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s + 4·4-s + 63·9-s + 24·12-s + 54·17-s − 374·23-s + 492·25-s + 594·27-s + 26·29-s + 252·36-s + 398·43-s − 661·49-s + 324·51-s + 2.47e3·53-s − 350·61-s − 64·64-s + 216·68-s − 2.24e3·69-s + 2.95e3·75-s + 3.05e3·79-s + 3.32e3·81-s + 156·87-s − 1.49e3·92-s + 1.96e3·100-s − 1.61e3·101-s − 5.15e3·103-s − 2.55e3·107-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1/2·4-s + 7/3·9-s + 0.577·12-s + 0.770·17-s − 3.39·23-s + 3.93·25-s + 4.23·27-s + 0.166·29-s + 7/6·36-s + 1.41·43-s − 1.92·49-s + 0.889·51-s + 6.40·53-s − 0.734·61-s − 1/8·64-s + 0.385·68-s − 3.91·69-s + 4.54·75-s + 4.35·79-s + 41/9·81-s + 0.192·87-s − 1.69·92-s + 1.96·100-s − 1.59·101-s − 4.92·103-s − 2.30·107-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$158172.$$ Root analytic conductor: $$4.46571$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$12.58630676$$ $$L(\frac12)$$ $$\approx$$ $$12.58630676$$ $$L(\frac{5}{2})$$ 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)$
bad2$C_2^2$ $$1 - p^{2} T^{2} + p^{4} T^{4}$$
13 $$1$$
good3$C_2^2$ $$( 1 - p T - 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} )^{2}$$
5$C_2^2$ $$( 1 - 246 T^{2} + p^{6} T^{4} )^{2}$$
7$C_2^3$ $$1 + 661 T^{2} + 319272 T^{4} + 661 p^{6} T^{6} + p^{12} T^{8}$$
11$C_2^3$ $$1 + 2493 T^{2} + 4443488 T^{4} + 2493 p^{6} T^{6} + p^{12} T^{8}$$
17$C_2^2$ $$( 1 - 27 T - 4184 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
19$C_2^3$ $$1 + 8093 T^{2} + 18450768 T^{4} + 8093 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2^2$ $$( 1 + 187 T + 22802 T^{2} + 187 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 13 T - 24220 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 48766 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2^3$ $$1 - 77623 T^{2} + 3459603720 T^{4} - 77623 p^{6} T^{6} + p^{12} T^{8}$$
41$C_2^3$ $$1 + 99817 T^{2} + 5213329248 T^{4} + 99817 p^{6} T^{6} + p^{12} T^{8}$$
43$C_2^2$ $$( 1 - 199 T - 39906 T^{2} - 199 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 57102 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2$ $$( 1 - 618 T + p^{3} T^{2} )^{4}$$
59$C_2^3$ $$1 + 169677 T^{2} - 13390249312 T^{4} + 169677 p^{6} T^{6} + p^{12} T^{8}$$
61$C_2^2$ $$( 1 + 175 T - 196356 T^{2} + 175 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$C_2^3$ $$1 - 65963 T^{2} - 86107264800 T^{4} - 65963 p^{6} T^{6} + p^{12} T^{8}$$
71$C_2^3$ $$1 + 709581 T^{2} + 375404911640 T^{4} + 709581 p^{6} T^{6} + p^{12} T^{8}$$
73$C_2^2$ $$( 1 - 725134 T^{2} + p^{6} T^{4} )^{2}$$
79$C_2$ $$( 1 - 764 T + p^{3} T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 607750 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^3$ $$1 + 326257 T^{2} - 390537660912 T^{4} + 326257 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^3$ $$1 + 193 p^{2} T^{2} + 27840 p^{4} T^{4} + 193 p^{8} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$