# Properties

 Degree $8$ Conductor $15752961$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 47·16-s − 500·25-s + 686·49-s + 2.96e3·67-s − 5.53e3·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
 L(s)  = 1 + 0.734·16-s − 4·25-s + 2·49-s + 5.39·67-s − 7.88·79-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$15752961$$    =    $$3^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{63} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 15752961,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.64626$$ $$L(\frac12)$$ $$\approx$$ $$1.64626$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
7$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
good2$C_2^3$ $$1 - 47 T^{4} + p^{12} T^{8}$$
5$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
11$C_2^3$ $$1 + 306322 T^{4} + p^{12} T^{8}$$
13$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
17$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
19$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
23$C_2^3$ $$1 + 220762978 T^{4} + p^{12} T^{8}$$
29$C_2^3$ $$1 - 739273358 T^{4} + p^{12} T^{8}$$
31$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 101194 T^{2} + p^{6} T^{4} )^{2}$$
41$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 126614 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
53$C_2^3$ $$1 - 41794002542 T^{4} + p^{12} T^{8}$$
59$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
61$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
67$C_2$ $$( 1 - 740 T + p^{3} T^{2} )^{4}$$
71$C_2^3$ $$1 - 197404987358 T^{4} + p^{12} T^{8}$$
73$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
79$C_2$ $$( 1 + 1384 T + p^{3} T^{2} )^{4}$$
83$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
89$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
97$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$