# Properties

 Label 8-1200e4-1.1-c2e4-0-21 Degree $8$ Conductor $2.074\times 10^{12}$ Sign $1$ Analytic cond. $1.14304\times 10^{6}$ Root an. cond. $5.71818$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 8·7-s + 4·9-s + 40·13-s − 32·19-s + 32·21-s + 4·27-s − 32·31-s + 88·37-s + 160·39-s + 56·43-s + 24·49-s − 128·57-s − 64·61-s + 32·63-s − 328·67-s − 200·73-s + 112·79-s + 31·81-s + 320·91-s − 128·93-s − 296·97-s + 296·103-s + 80·109-s + 352·111-s + 160·117-s + 340·121-s + ⋯
 L(s)  = 1 + 4/3·3-s + 8/7·7-s + 4/9·9-s + 3.07·13-s − 1.68·19-s + 1.52·21-s + 4/27·27-s − 1.03·31-s + 2.37·37-s + 4.10·39-s + 1.30·43-s + 0.489·49-s − 2.24·57-s − 1.04·61-s + 0.507·63-s − 4.89·67-s − 2.73·73-s + 1.41·79-s + 0.382·81-s + 3.51·91-s − 1.37·93-s − 3.05·97-s + 2.87·103-s + 0.733·109-s + 3.17·111-s + 1.36·117-s + 2.80·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{4} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$1.14304\times 10^{6}$$ Root analytic conductor: $$5.71818$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$8.109971645$$ $$L(\frac12)$$ $$\approx$$ $$8.109971645$$ $$L(2)$$ 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)$
bad2 $$1$$
3$C_2^2$ $$1 - 4 T + 4 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4}$$
5 $$1$$
good7$D_{4}$ $$( 1 - 4 T + 12 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 170 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{4}$$
17$D_4\times C_2$ $$1 - 220 T^{2} - 28218 T^{4} - 220 p^{4} T^{6} + p^{8} T^{8}$$
19$D_{4}$ $$( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 1720 T^{2} + 1286322 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8}$$
29$C_2^2$ $$( 1 - 962 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2$ $$( 1 + 8 T + p^{2} T^{2} )^{4}$$
37$D_{4}$ $$( 1 - 44 T + 1782 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 4060 T^{2} + 8942982 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 - 28 T + 3084 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 1064 T^{2} + 1942386 T^{4} + 1064 p^{4} T^{6} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 10300 T^{2} + 42096102 T^{4} - 10300 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 7300 T^{2} + 30092262 T^{4} - 7300 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 + 32 T + 6258 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 164 T + 14892 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 15124 T^{2} + 102823206 T^{4} - 15124 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 100 T + 11718 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 56 T + 12906 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 26872 T^{2} + 275326098 T^{4} - 26872 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 27940 T^{2} + 317327622 T^{4} - 27940 p^{4} T^{6} + p^{8} T^{8}$$
97$D_{4}$ $$( 1 + 148 T + 22854 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$