Properties

 Label 8-2160e4-1.1-c2e4-0-1 Degree $8$ Conductor $2.177\times 10^{13}$ Sign $1$ Analytic cond. $1.19992\times 10^{7}$ Root an. cond. $7.67174$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 20·7-s − 28·13-s + 4·19-s − 10·25-s + 28·31-s − 64·37-s − 116·43-s + 144·49-s − 4·61-s − 56·67-s − 28·73-s + 268·79-s + 560·91-s + 8·97-s + 400·103-s + 140·109-s + 376·121-s + 127-s + 131-s − 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 − 2.85·7-s − 2.15·13-s + 4/19·19-s − 2/5·25-s + 0.903·31-s − 1.72·37-s − 2.69·43-s + 2.93·49-s − 0.0655·61-s − 0.835·67-s − 0.383·73-s + 3.39·79-s + 6.15·91-s + 8/97·97-s + 3.88·103-s + 1.28·109-s + 3.10·121-s + 0.00787·127-s + 0.00763·131-s − 0.601·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{12} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$1.19992\times 10^{7}$$ Root analytic conductor: $$7.67174$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2648810944$$ $$L(\frac12)$$ $$\approx$$ $$0.2648810944$$ $$L(2)$$ 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 $$1$$
5$C_2$ $$( 1 + p T^{2} )^{2}$$
good7$D_{4}$ $$( 1 + 10 T + 78 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 376 T^{2} + 63006 T^{4} - 376 p^{4} T^{6} + p^{8} T^{8}$$
13$D_{4}$ $$( 1 + 14 T + 342 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 353 T^{2} + p^{4} T^{4} )^{2}$$
19$D_{4}$ $$( 1 - 2 T + 543 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 38 p T^{2} + 433131 T^{4} - 38 p^{5} T^{6} + p^{8} T^{8}$$
29$D_4\times C_2$ $$1 - 3256 T^{2} + 4063326 T^{4} - 3256 p^{4} T^{6} + p^{8} T^{8}$$
31$D_{4}$ $$( 1 - 14 T + 1791 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 32 T + 1374 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 4312 T^{2} + 9935358 T^{4} - 4312 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 + 58 T + 2334 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 8188 T^{2} + 26416518 T^{4} - 8188 p^{4} T^{6} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 2218 T^{2} + 16848843 T^{4} - 2218 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 7912 T^{2} + 35671038 T^{4} - 7912 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 + 2 T + 963 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$C_2$ $$( 1 + 14 T + p^{2} T^{2} )^{4}$$
71$D_4\times C_2$ $$1 - 11704 T^{2} + 67208766 T^{4} - 11704 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 14 T + 10302 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 134 T + 10491 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 2338 T^{2} + 46102083 T^{4} - 2338 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 5512 T^{2} + 128866398 T^{4} - 5512 p^{4} T^{6} + p^{8} T^{8}$$
97$D_{4}$ $$( 1 - 4 T + 18642 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$