Properties

 Label 8-2160e4-1.1-c2e4-0-11 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 + 14·7-s + 66·19-s − 10·25-s − 70·31-s + 14·37-s + 30·43-s − 49-s + 66·61-s + 74·67-s + 210·73-s + 212·79-s − 202·97-s − 110·103-s − 304·109-s + 467·121-s + 127-s + 131-s + 924·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 386·169-s + 173-s + ⋯
 L(s)  = 1 + 2·7-s + 3.47·19-s − 2/5·25-s − 2.25·31-s + 0.378·37-s + 0.697·43-s − 0.0204·49-s + 1.08·61-s + 1.10·67-s + 2.87·73-s + 2.68·79-s − 2.08·97-s − 1.06·103-s − 2.78·109-s + 3.85·121-s + 0.00787·127-s + 0.00763·131-s + 6.94·133-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 − 2.28·169-s + 0.00578·173-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

 $$L(\frac{3}{2})$$ $$\approx$$ $$11.29210183$$ $$L(\frac12)$$ $$\approx$$ $$11.29210183$$ $$L(2)$$ 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$D_{4}$ $$( 1 - p T + 74 T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 467 T^{2} + 83768 T^{4} - 467 p^{4} T^{6} + p^{8} T^{8}$$
13$C_2^2$ $$( 1 + 193 T^{2} + p^{4} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 383 T^{2} + 159308 T^{4} - 383 p^{4} T^{6} + p^{8} T^{8}$$
19$D_{4}$ $$( 1 - 33 T + 958 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 163 T^{2} + 328488 T^{4} - 163 p^{4} T^{6} + p^{8} T^{8}$$
29$D_4\times C_2$ $$1 - 611 T^{2} + 766616 T^{4} - 611 p^{4} T^{6} + p^{8} T^{8}$$
31$D_{4}$ $$( 1 + 35 T + 2192 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 7 T + 2424 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 2012 T^{2} + 2951558 T^{4} - 2012 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 - 15 T - 632 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 5519 T^{2} + 14632796 T^{4} - 5519 p^{4} T^{6} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 10844 T^{2} + 45141926 T^{4} - 10844 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 4064 T^{2} + 16169246 T^{4} - 4064 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 - 33 T + 6808 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_{4}$ $$( 1 - 37 T - 1156 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 12896 T^{2} + 90645566 T^{4} - 12896 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 - 105 T + 11638 T^{2} - 105 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 106 T + 12971 T^{2} - 106 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 3004 T^{2} - 11069274 T^{4} - 3004 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 17572 T^{2} + 154570758 T^{4} - 17572 p^{4} T^{6} + p^{8} T^{8}$$
97$D_{4}$ $$( 1 + 101 T + 13212 T^{2} + 101 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$