# Properties

 Label 8-48e4-1.1-c10e4-0-0 Degree $8$ Conductor $5308416$ Sign $1$ Analytic cond. $865041.$ Root an. cond. $5.52242$ Motivic weight $10$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 7.56e3·5-s − 3.93e4·9-s − 5.36e5·13-s − 3.85e6·17-s + 2.86e7·25-s + 1.12e7·29-s + 3.05e7·37-s − 2.37e8·41-s + 2.97e8·45-s + 1.07e8·49-s + 8.35e8·53-s + 2.61e9·61-s + 4.05e9·65-s − 2.37e8·73-s + 1.16e9·81-s + 2.91e10·85-s − 1.11e10·89-s + 9.66e9·97-s − 3.73e10·101-s + 1.10e10·109-s − 7.13e10·113-s + 2.11e10·117-s + 6.79e10·121-s − 1.28e11·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 2.41·5-s − 2/3·9-s − 1.44·13-s − 2.71·17-s + 2.93·25-s + 0.549·29-s + 0.439·37-s − 2.05·41-s + 1.61·45-s + 0.382·49-s + 1.99·53-s + 3.09·61-s + 3.49·65-s − 0.114·73-s + 1/3·81-s + 6.57·85-s − 2.00·89-s + 1.12·97-s − 3.55·101-s + 0.717·109-s − 3.87·113-s + 0.962·117-s + 2.62·121-s − 4.20·125-s − 1.33·145-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5308416$$    =    $$2^{16} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$865041.$$ Root analytic conductor: $$5.52242$$ Motivic weight: $$10$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5308416,\ (\ :5, 5, 5, 5),\ 1)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.01266944810$$ $$L(\frac12)$$ $$\approx$$ $$0.01266944810$$ $$L(6)$$ 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$C_2$ $$( 1 + p^{9} T^{2} )^{2}$$
good5$D_{4}$ $$( 1 + 756 p T + 1424446 p T^{2} + 756 p^{11} T^{3} + p^{20} T^{4} )^{2}$$
7$D_4\times C_2$ $$1 - 107909092 T^{2} + 515752461638502 p^{2} T^{4} - 107909092 p^{20} T^{6} + p^{40} T^{8}$$
11$D_4\times C_2$ $$1 - 67987804420 T^{2} +$$$$23\!\cdots\!82$$$$T^{4} - 67987804420 p^{20} T^{6} + p^{40} T^{8}$$
13$D_{4}$ $$( 1 + 268156 T + 247092947862 T^{2} + 268156 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
17$D_{4}$ $$( 1 + 1928556 T + 4674862871462 T^{2} + 1928556 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 11396009075908 T^{2} +$$$$67\!\cdots\!98$$$$T^{4} - 11396009075908 p^{20} T^{6} + p^{40} T^{8}$$
23$D_4\times C_2$ $$1 - 57560728155460 T^{2} +$$$$42\!\cdots\!22$$$$T^{4} - 57560728155460 p^{20} T^{6} + p^{40} T^{8}$$
29$D_{4}$ $$( 1 - 5639436 T + 551257283207606 T^{2} - 5639436 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 1785509547336484 T^{2} +$$$$18\!\cdots\!66$$$$T^{4} - 1785509547336484 p^{20} T^{6} + p^{40} T^{8}$$
37$D_{4}$ $$( 1 - 15255220 T + 9227793694532118 T^{2} - 15255220 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
41$D_{4}$ $$( 1 + 118854540 T + 28110027532085702 T^{2} + 118854540 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 3337921746848260 T^{2} +$$$$68\!\cdots\!22$$$$T^{4} - 3337921746848260 p^{20} T^{6} + p^{40} T^{8}$$
47$D_4\times C_2$ $$1 - 22958275505088772 T^{2} -$$$$31\!\cdots\!82$$$$T^{4} - 22958275505088772 p^{20} T^{6} + p^{40} T^{8}$$
53$D_{4}$ $$( 1 - 417942828 T + 380158218111049814 T^{2} - 417942828 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 1717674283247984260 T^{2} +$$$$12\!\cdots\!82$$$$T^{4} - 1717674283247984260 p^{20} T^{6} + p^{40} T^{8}$$
61$D_{4}$ $$( 1 - 1306619188 T + 1842940593015440118 T^{2} - 1306619188 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 3067696319988122692 T^{2} +$$$$51\!\cdots\!98$$$$T^{4} - 3067696319988122692 p^{20} T^{6} + p^{40} T^{8}$$
71$D_4\times C_2$ $$1 - 9494363384388213700 T^{2} +$$$$43\!\cdots\!22$$$$T^{4} - 9494363384388213700 p^{20} T^{6} + p^{40} T^{8}$$
73$D_{4}$ $$( 1 + 118687324 T + 7332861243362857062 T^{2} + 118687324 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 15414374288754462628 T^{2} +$$$$19\!\cdots\!78$$$$T^{4} - 15414374288754462628 p^{20} T^{6} + p^{40} T^{8}$$
83$D_4\times C_2$ $$1 - 27382880992584417220 T^{2} +$$$$47\!\cdots\!22$$$$T^{4} - 27382880992584417220 p^{20} T^{6} + p^{40} T^{8}$$
89$D_{4}$ $$( 1 + 5592224988 T + 68730968898102971558 T^{2} + 5592224988 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
97$D_{4}$ $$( 1 - 4834774532 T - 44713135597567105146 T^{2} - 4834774532 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$