# Properties

 Label 8-210e4-1.1-c7e4-0-0 Degree $8$ Conductor $1944810000$ Sign $1$ Analytic cond. $1.85198\times 10^{7}$ Root an. cond. $8.09943$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s − 54·3-s + 64·4-s + 250·5-s + 864·6-s − 77·7-s + 1.02e3·8-s + 729·9-s − 4.00e3·10-s + 5.32e3·11-s − 3.45e3·12-s − 1.37e4·13-s + 1.23e3·14-s − 1.35e4·15-s − 1.63e4·16-s − 2.17e4·17-s − 1.16e4·18-s − 4.46e4·19-s + 1.60e4·20-s + 4.15e3·21-s − 8.52e4·22-s + 4.61e4·23-s − 5.52e4·24-s + 1.56e4·25-s + 2.19e5·26-s + 3.93e4·27-s − 4.92e3·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.0848·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s − 0.577·12-s − 1.73·13-s + 0.119·14-s − 1.03·15-s − 16-s − 1.07·17-s − 0.471·18-s − 1.49·19-s + 0.447·20-s + 0.0979·21-s − 1.70·22-s + 0.790·23-s − 0.816·24-s + 1/5·25-s + 2.44·26-s + 0.384·27-s − 0.0424·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$1.85198\times 10^{7}$$ Root analytic conductor: $$8.09943$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{210} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.4328965164$$ $$L(\frac12)$$ $$\approx$$ $$0.4328965164$$ $$L(\frac{9}{2})$$ 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$C_2$ $$( 1 + p^{3} T + p^{6} T^{2} )^{2}$$
3$C_2$ $$( 1 + p^{3} T + p^{6} T^{2} )^{2}$$
5$C_2$ $$( 1 - p^{3} T + p^{6} T^{2} )^{2}$$
7$C_2^2$ $$1 + 11 p T - 16686 p^{2} T^{2} + 11 p^{8} T^{3} + p^{14} T^{4}$$
good11$D_4\times C_2$ $$1 - 5325 T + 8917079 T^{2} + 104028113700 T^{3} - 554463676176000 T^{4} + 104028113700 p^{7} T^{5} + 8917079 p^{14} T^{6} - 5325 p^{21} T^{7} + p^{28} T^{8}$$
13$D_{4}$ $$( 1 + 6860 T + 30763125 T^{2} + 6860 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 21786 T - 245466430 T^{2} - 2191260280320 T^{3} + 155401787483141439 T^{4} - 2191260280320 p^{7} T^{5} - 245466430 p^{14} T^{6} + 21786 p^{21} T^{7} + p^{28} T^{8}$$
19$D_4\times C_2$ $$1 + 44692 T + 155005759 T^{2} + 128490974836 p T^{3} + 2243546510130328 p^{2} T^{4} + 128490974836 p^{8} T^{5} + 155005759 p^{14} T^{6} + 44692 p^{21} T^{7} + p^{28} T^{8}$$
23$D_4\times C_2$ $$1 - 46143 T - 5060519551 T^{2} - 17536421326158 T^{3} + 32849418093168952488 T^{4} - 17536421326158 p^{7} T^{5} - 5060519551 p^{14} T^{6} - 46143 p^{21} T^{7} + p^{28} T^{8}$$
29$D_{4}$ $$( 1 - 240654 T + 46924664026 T^{2} - 240654 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 341927 T + 33133884097 T^{2} - 9832097553266270 T^{3} +$$$$30\!\cdots\!84$$$$T^{4} - 9832097553266270 p^{7} T^{5} + 33133884097 p^{14} T^{6} - 341927 p^{21} T^{7} + p^{28} T^{8}$$
37$D_4\times C_2$ $$1 + 258856 T - 138288065633 T^{2} + 3994339660102168 T^{3} +$$$$26\!\cdots\!84$$$$T^{4} + 3994339660102168 p^{7} T^{5} - 138288065633 p^{14} T^{6} + 258856 p^{21} T^{7} + p^{28} T^{8}$$
41$D_{4}$ $$( 1 + 320979 T + 390391764850 T^{2} + 320979 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
43$D_{4}$ $$( 1 + 990095 T + 710946547038 T^{2} + 990095 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 1722897 T + 1230732895613 T^{2} + 1248057862170194790 T^{3} +$$$$12\!\cdots\!80$$$$T^{4} + 1248057862170194790 p^{7} T^{5} + 1230732895613 p^{14} T^{6} + 1722897 p^{21} T^{7} + p^{28} T^{8}$$
53$D_4\times C_2$ $$1 + 2994681 T + 4391722702919 T^{2} + 6669062677748535408 T^{3} +$$$$90\!\cdots\!38$$$$T^{4} + 6669062677748535408 p^{7} T^{5} + 4391722702919 p^{14} T^{6} + 2994681 p^{21} T^{7} + p^{28} T^{8}$$
59$D_4\times C_2$ $$1 - 14550 p T + 610437743558 T^{2} + 70579202720626800 p T^{3} -$$$$80\!\cdots\!97$$$$T^{4} + 70579202720626800 p^{8} T^{5} + 610437743558 p^{14} T^{6} - 14550 p^{22} T^{7} + p^{28} T^{8}$$
61$D_4\times C_2$ $$1 - 3920558 T + 5365393559962 T^{2} - 14584067235347242880 T^{3} +$$$$41\!\cdots\!19$$$$T^{4} - 14584067235347242880 p^{7} T^{5} + 5365393559962 p^{14} T^{6} - 3920558 p^{21} T^{7} + p^{28} T^{8}$$
67$D_4\times C_2$ $$1 - 4109921 T + 666683047435 T^{2} - 16864421188932515360 T^{3} +$$$$12\!\cdots\!64$$$$T^{4} - 16864421188932515360 p^{7} T^{5} + 666683047435 p^{14} T^{6} - 4109921 p^{21} T^{7} + p^{28} T^{8}$$
71$D_{4}$ $$( 1 + 6286110 T + 344627063426 p T^{2} + 6286110 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 4115065 T - 5172409247975 T^{2} + 46797194075350390 T^{3} +$$$$17\!\cdots\!66$$$$T^{4} + 46797194075350390 p^{7} T^{5} - 5172409247975 p^{14} T^{6} + 4115065 p^{21} T^{7} + p^{28} T^{8}$$
79$D_4\times C_2$ $$1 - 1753115 T - 22151757036269 T^{2} + 23110699233244746760 T^{3} +$$$$20\!\cdots\!80$$$$T^{4} + 23110699233244746760 p^{7} T^{5} - 22151757036269 p^{14} T^{6} - 1753115 p^{21} T^{7} + p^{28} T^{8}$$
83$D_{4}$ $$( 1 - 15542832 T + 114172035982006 T^{2} - 15542832 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 5883408 T - 55762019151694 T^{2} - 11259896007487780800 T^{3} +$$$$46\!\cdots\!67$$$$T^{4} - 11259896007487780800 p^{7} T^{5} - 55762019151694 p^{14} T^{6} - 5883408 p^{21} T^{7} + p^{28} T^{8}$$
97$D_{4}$ $$( 1 + 10759946 T + 151301298759714 T^{2} + 10759946 p^{7} T^{3} + p^{14} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$