# Properties

 Label 8-384e4-1.1-c3e4-0-7 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $263505.$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 32·7-s − 18·9-s − 72·17-s + 512·23-s + 52·25-s + 160·31-s + 872·41-s + 448·47-s − 316·49-s − 576·63-s + 4.09e3·71-s − 1.32e3·73-s − 992·79-s + 243·81-s − 1.06e3·89-s − 2.44e3·97-s + 3.42e3·103-s + 456·113-s − 2.30e3·119-s − 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.29e3·153-s + ⋯
 L(s)  = 1 + 1.72·7-s − 2/3·9-s − 1.02·17-s + 4.64·23-s + 0.415·25-s + 0.926·31-s + 3.32·41-s + 1.39·47-s − 0.921·49-s − 1.15·63-s + 6.84·71-s − 2.11·73-s − 1.41·79-s + 1/3·81-s − 1.26·89-s − 2.55·97-s + 3.27·103-s + 0.379·113-s − 1.77·119-s − 1.02·121-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 + 0.684·153-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$263505.$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$7.978498188$$ $$L(\frac12)$$ $$\approx$$ $$7.978498188$$ $$L(\frac{5}{2})$$ 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 $$1$$
3$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 - 52 T^{2} + 18614 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8}$$
7$D_{4}$ $$( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 + 124 p T^{2} + 3795254 T^{4} + 124 p^{7} T^{6} + p^{12} T^{8}$$
13$D_4\times C_2$ $$1 - 4532 T^{2} + 10475286 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8}$$
17$D_{4}$ $$( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 11532 T^{2} + 65783830 T^{4} - 11532 p^{6} T^{6} + p^{12} T^{8}$$
23$D_{4}$ $$( 1 - 256 T + 39886 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 52308 T^{2} + 1484422166 T^{4} - 52308 p^{6} T^{6} + p^{12} T^{8}$$
31$D_{4}$ $$( 1 - 80 T + 26030 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 186708 T^{2} + 13784867446 T^{4} - 186708 p^{6} T^{6} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 - 436 T + 102166 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 57900 T^{2} + 11793718966 T^{4} - 57900 p^{6} T^{6} + p^{12} T^{8}$$
47$D_{4}$ $$( 1 - 224 T + 179422 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 442740 T^{2} + 87795274358 T^{4} - 442740 p^{6} T^{6} + p^{12} T^{8}$$
59$C_2^2$ $$( 1 - 305782 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 348986 T^{2} + p^{6} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 140620 T^{2} + 86210675286 T^{4} - 140620 p^{6} T^{6} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 - 2048 T + 1763566 T^{2} - 2048 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 660 T + 407702 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 496 T + 323534 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 1659916 T^{2} + 1244512983830 T^{4} - 1659916 p^{6} T^{6} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 + 532 T + 1467382 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$