# Properties

 Label 8-768e4-1.1-c1e4-0-15 Degree $8$ Conductor $347892350976$ Sign $1$ Analytic cond. $1414.33$ Root an. cond. $2.47639$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·13-s + 24·17-s + 24·29-s + 20·37-s + 24·49-s − 24·53-s + 20·61-s − 81-s − 48·97-s − 24·101-s − 52·109-s + 24·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 1.10·13-s + 5.82·17-s + 4.45·29-s + 3.28·37-s + 24/7·49-s − 3.29·53-s + 2.56·61-s − 1/9·81-s − 4.87·97-s − 2.38·101-s − 4.98·109-s + 2.25·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{32} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$1414.33$$ Root analytic conductor: $$2.47639$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{768} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.105631884$$ $$L(\frac12)$$ $$\approx$$ $$6.105631884$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2^2$ $$1 + T^{4}$$
good5$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 12 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
19$C_2^3$ $$1 - 718 T^{4} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 60 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^3$ $$1 - 1198 T^{4} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + 22 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 4946 T^{4} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 - p T^{2} )^{4}$$
79$C_2^2$ $$( 1 + 156 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^3$ $$1 - 13294 T^{4} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 142 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 12 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$