Properties

 Label 8-18e4-1.1-c3e4-0-0 Degree $8$ Conductor $104976$ Sign $1$ Analytic cond. $1.27219$ Root an. cond. $1.03055$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 4·2-s + 3·3-s + 4·4-s + 9·5-s + 12·6-s − 19·7-s − 16·8-s − 21·9-s + 36·10-s + 24·11-s + 12·12-s − 61·13-s − 76·14-s + 27·15-s − 64·16-s + 6·17-s − 84·18-s + 266·19-s + 36·20-s − 57·21-s + 96·22-s − 69·23-s − 48·24-s + 34·25-s − 244·26-s − 72·27-s − 76·28-s + ⋯
 L(s)  = 1 + 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.804·5-s + 0.816·6-s − 1.02·7-s − 0.707·8-s − 7/9·9-s + 1.13·10-s + 0.657·11-s + 0.288·12-s − 1.30·13-s − 1.45·14-s + 0.464·15-s − 16-s + 0.0856·17-s − 1.09·18-s + 3.21·19-s + 0.402·20-s − 0.592·21-s + 0.930·22-s − 0.625·23-s − 0.408·24-s + 0.271·25-s − 1.84·26-s − 0.513·27-s − 0.512·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$104976$$    =    $$2^{4} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$1.27219$$ Root analytic conductor: $$1.03055$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{18} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 104976,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.221061581$$ $$L(\frac12)$$ $$\approx$$ $$2.221061581$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
3$C_2^2$ $$1 - p T + 10 p T^{2} - p^{4} T^{3} + p^{6} T^{4}$$
good5$D_4\times C_2$ $$1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 1944 p^{3} T^{5} + 47 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8}$$
7$D_4\times C_2$ $$1 + 19 T - 179 T^{2} - 2774 T^{3} + 50128 T^{4} - 2774 p^{3} T^{5} - 179 p^{6} T^{6} + 19 p^{9} T^{7} + p^{12} T^{8}$$
11$D_4\times C_2$ $$1 - 24 T - 1285 T^{2} + 19224 T^{3} + 925104 T^{4} + 19224 p^{3} T^{5} - 1285 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8}$$
13$D_4\times C_2$ $$1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 42334 p^{3} T^{5} - 1367 p^{6} T^{6} + 61 p^{9} T^{7} + p^{12} T^{8}$$
17$D_{4}$ $$( 1 - 3 T + 9592 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
19$D_{4}$ $$( 1 - 7 p T + 12234 T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 + 3 p T - 20527 T^{2} + 2862 p T^{3} + 433519968 T^{4} + 2862 p^{4} T^{5} - 20527 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8}$$
29$D_4\times C_2$ $$1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 584442 p^{3} T^{5} + 4925 p^{6} T^{6} + 237 p^{9} T^{7} + p^{12} T^{8}$$
31$D_4\times C_2$ $$1 + 211 T - 24065 T^{2} + 1899844 T^{3} + 2490210604 T^{4} + 1899844 p^{3} T^{5} - 24065 p^{6} T^{6} + 211 p^{9} T^{7} + p^{12} T^{8}$$
37$D_{4}$ $$( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 25183548 p^{3} T^{5} + 27371 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8}$$
43$D_4\times C_2$ $$1 - 2 p T - 17387 T^{2} + 268462 p T^{3} - 6295199732 T^{4} + 268462 p^{4} T^{5} - 17387 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8}$$
47$D_4\times C_2$ $$1 + 483 T + 20477 T^{2} + 2495178 T^{3} + 10288968168 T^{4} + 2495178 p^{3} T^{5} + 20477 p^{6} T^{6} + 483 p^{9} T^{7} + p^{12} T^{8}$$
53$D_{4}$ $$( 1 - 150 T + 257074 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 + 168 T - 388645 T^{2} + 1026648 T^{3} + 125802612624 T^{4} + 1026648 p^{3} T^{5} - 388645 p^{6} T^{6} + 168 p^{9} T^{7} + p^{12} T^{8}$$
61$D_4\times C_2$ $$1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 232819256 p^{3} T^{5} + 424495 p^{6} T^{6} - 1049 p^{9} T^{7} + p^{12} T^{8}$$
67$D_4\times C_2$ $$1 - 1166 T + 452161 T^{2} - 356643254 T^{3} + 324003162628 T^{4} - 356643254 p^{3} T^{5} + 452161 p^{6} T^{6} - 1166 p^{9} T^{7} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 312 T + 498238 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 + 349 T - 854801 T^{2} - 3307124 T^{3} + 650611367644 T^{4} - 3307124 p^{3} T^{5} - 854801 p^{6} T^{6} + 349 p^{9} T^{7} + p^{12} T^{8}$$
83$D_4\times C_2$ $$1 + 1221 T + 78743 T^{2} + 327867804 T^{3} + 814636885368 T^{4} + 327867804 p^{3} T^{5} + 78743 p^{6} T^{6} + 1221 p^{9} T^{7} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 + 492 T + 1092454 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 14111872 p^{3} T^{5} - 1698713 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$