# Properties

 Label 8-63e4-1.1-c5e4-0-2 Degree $8$ Conductor $15752961$ Sign $1$ Analytic cond. $10423.2$ Root an. cond. $3.17870$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 28·4-s − 38·5-s − 168·7-s + 32·8-s − 76·10-s + 424·11-s − 1.84e3·13-s − 336·14-s + 960·16-s − 2.34e3·17-s + 360·19-s − 1.06e3·20-s + 848·22-s − 12·23-s + 2.91e3·25-s − 3.69e3·26-s − 4.70e3·28-s + 1.41e4·29-s − 3.54e3·31-s − 896·32-s − 4.69e3·34-s + 6.38e3·35-s − 1.10e4·37-s + 720·38-s − 1.21e3·40-s − 7.00e3·41-s + ⋯
 L(s)  = 1 + 0.353·2-s + 7/8·4-s − 0.679·5-s − 1.29·7-s + 0.176·8-s − 0.240·10-s + 1.05·11-s − 3.03·13-s − 0.458·14-s + 0.937·16-s − 1.96·17-s + 0.228·19-s − 0.594·20-s + 0.373·22-s − 0.00473·23-s + 0.931·25-s − 1.07·26-s − 1.13·28-s + 3.11·29-s − 0.663·31-s − 0.154·32-s − 0.696·34-s + 0.880·35-s − 1.33·37-s + 0.0808·38-s − 0.120·40-s − 0.650·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$15752961$$    =    $$3^{8} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$10423.2$$ Root analytic conductor: $$3.17870$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 15752961,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.5627607853$$ $$L(\frac12)$$ $$\approx$$ $$0.5627607853$$ $$L(\frac{7}{2})$$ 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$C_2^2$ $$1 + 24 p T + 34 p^{3} T^{2} + 24 p^{6} T^{3} + p^{10} T^{4}$$
good2$D_4\times C_2$ $$1 - p T - 3 p^{3} T^{2} + 9 p^{3} T^{3} - 23 p^{4} T^{4} + 9 p^{8} T^{5} - 3 p^{13} T^{6} - p^{16} T^{7} + p^{20} T^{8}$$
5$D_4\times C_2$ $$1 + 38 T - 1467 T^{2} - 126882 T^{3} - 5804204 T^{4} - 126882 p^{5} T^{5} - 1467 p^{10} T^{6} + 38 p^{15} T^{7} + p^{20} T^{8}$$
11$D_4\times C_2$ $$1 - 424 T - 167697 T^{2} - 10757304 T^{3} + 65846956552 T^{4} - 10757304 p^{5} T^{5} - 167697 p^{10} T^{6} - 424 p^{15} T^{7} + p^{20} T^{8}$$
13$D_{4}$ $$( 1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 138 p T + 1932761 T^{2} + 100911258 p T^{3} + 2921236022964 T^{4} + 100911258 p^{6} T^{5} + 1932761 p^{10} T^{6} + 138 p^{16} T^{7} + p^{20} T^{8}$$
19$D_4\times C_2$ $$1 - 360 T - 2016025 T^{2} + 1010366280 T^{3} - 1848262047576 T^{4} + 1010366280 p^{5} T^{5} - 2016025 p^{10} T^{6} - 360 p^{15} T^{7} + p^{20} T^{8}$$
23$D_4\times C_2$ $$1 + 12 T - 12696421 T^{2} - 2113452 T^{3} + 119775324752184 T^{4} - 2113452 p^{5} T^{5} - 12696421 p^{10} T^{6} + 12 p^{15} T^{7} + p^{20} T^{8}$$
29$D_{4}$ $$( 1 - 7052 T + 35324974 T^{2} - 7052 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 3548 T - 28901749 T^{2} - 55945747452 T^{3} + 541403754912104 T^{4} - 55945747452 p^{5} T^{5} - 28901749 p^{10} T^{6} + 3548 p^{15} T^{7} + p^{20} T^{8}$$
37$D_4\times C_2$ $$1 + 11090 T - 23355139 T^{2} + 84897554250 T^{3} + 8079277710681572 T^{4} + 84897554250 p^{5} T^{5} - 23355139 p^{10} T^{6} + 11090 p^{15} T^{7} + p^{20} T^{8}$$
41$D_{4}$ $$( 1 + 3500 T + 206898214 T^{2} + 3500 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
43$D_{4}$ $$( 1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 22956 T + 35452763 T^{2} + 753763910004 T^{3} + 68138105036084328 T^{4} + 753763910004 p^{5} T^{5} + 35452763 p^{10} T^{6} + 22956 p^{15} T^{7} + p^{20} T^{8}$$
53$D_4\times C_2$ $$1 - 3042 T - 707161075 T^{2} + 364967439174 T^{3} + 334492868775797220 T^{4} + 364967439174 p^{5} T^{5} - 707161075 p^{10} T^{6} - 3042 p^{15} T^{7} + p^{20} T^{8}$$
59$D_4\times C_2$ $$1 + 65808 T + 1828603607 T^{2} + 70562013287472 T^{3} + 2653216336714668312 T^{4} + 70562013287472 p^{5} T^{5} + 1828603607 p^{10} T^{6} + 65808 p^{15} T^{7} + p^{20} T^{8}$$
61$D_4\times C_2$ $$1 - 42486 T + 303064037 T^{2} + 7953228077298 T^{3} + 18102385363935684 T^{4} + 7953228077298 p^{5} T^{5} + 303064037 p^{10} T^{6} - 42486 p^{15} T^{7} + p^{20} T^{8}$$
67$D_4\times C_2$ $$1 + 42312 T - 1326272449 T^{2} + 17615652522648 T^{3} + 5473083141138317592 T^{4} + 17615652522648 p^{5} T^{5} - 1326272449 p^{10} T^{6} + 42312 p^{15} T^{7} + p^{20} T^{8}$$
71$D_{4}$ $$( 1 - 2208 T + 3433192846 T^{2} - 2208 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 50506 T - 222184951 T^{2} + 69349899662694 T^{3} - 1895976700185808828 T^{4} + 69349899662694 p^{5} T^{5} - 222184951 p^{10} T^{6} - 50506 p^{15} T^{7} + p^{20} T^{8}$$
79$D_4\times C_2$ $$1 + 9004 T - 5095520573 T^{2} - 8801591961836 T^{3} + 17079371584250245336 T^{4} - 8801591961836 p^{5} T^{5} - 5095520573 p^{10} T^{6} + 9004 p^{15} T^{7} + p^{20} T^{8}$$
83$D_{4}$ $$( 1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 26666 T - 7396026999 T^{2} - 81625061802438 T^{3} + 30572703729886615780 T^{4} - 81625061802438 p^{5} T^{5} - 7396026999 p^{10} T^{6} + 26666 p^{15} T^{7} + p^{20} T^{8}$$
97$D_{4}$ $$( 1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$