# Properties

 Label 8-195e4-1.1-c2e4-0-3 Degree $8$ Conductor $1445900625$ Sign $1$ Analytic cond. $797.037$ Root an. cond. $2.30507$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·3-s + 32·9-s − 32·16-s + 88·19-s + 48·25-s + 72·27-s − 52·31-s + 68·43-s − 256·48-s + 146·49-s + 704·57-s + 268·61-s − 400·67-s − 304·73-s + 384·75-s + 47·81-s − 416·93-s − 100·97-s + 4·103-s + 396·109-s + 127-s + 544·129-s + 131-s + 137-s + 139-s − 1.02e3·144-s + 1.16e3·147-s + ⋯
 L(s)  = 1 + 8/3·3-s + 32/9·9-s − 2·16-s + 4.63·19-s + 1.91·25-s + 8/3·27-s − 1.67·31-s + 1.58·43-s − 5.33·48-s + 2.97·49-s + 12.3·57-s + 4.39·61-s − 5.97·67-s − 4.16·73-s + 5.11·75-s + 0.580·81-s − 4.47·93-s − 1.03·97-s + 0.0388·103-s + 3.63·109-s + 0.00787·127-s + 4.21·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.11·144-s + 7.94·147-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$797.037$$ Root analytic conductor: $$2.30507$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$9.347535282$$ $$L(\frac12)$$ $$\approx$$ $$9.347535282$$ $$L(2)$$ 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)$
bad3$C_2^2$ $$1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4}$$
5$C_2^2$ $$1 - 48 T^{2} + p^{4} T^{4}$$
13$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good2$C_2^2$ $$( 1 + p^{4} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 73 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^3$ $$1 - 23953 T^{4} + p^{8} T^{8}$$
17$C_2^3$ $$1 - 119953 T^{4} + p^{8} T^{8}$$
19$C_2^2$ $$( 1 - 44 T + 968 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_2^3$ $$1 - 178993 T^{4} + p^{8} T^{8}$$
29$C_2^2$ $$( 1 + 224 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 26 T + 338 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 2513 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2^3$ $$1 - 1084753 T^{4} + p^{8} T^{8}$$
43$C_2^2$ $$( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 784 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^3$ $$1 + 13910639 T^{4} + p^{8} T^{8}$$
59$C_2^3$ $$1 - 18768478 T^{4} + p^{8} T^{8}$$
61$C_2$ $$( 1 - 67 T + p^{2} T^{2} )^{4}$$
67$C_2$ $$( 1 + 100 T + p^{2} T^{2} )^{4}$$
71$C_2^3$ $$1 + 2073167 T^{4} + p^{8} T^{8}$$
73$C_2$ $$( 1 + 76 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 4561 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 5066 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^3$ $$1 + 123934367 T^{4} + p^{8} T^{8}$$
97$C_2$ $$( 1 + 25 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$