Properties

 Label 8-720e4-1.1-c3e4-0-8 Degree $8$ Conductor $268738560000$ Sign $1$ Analytic cond. $3.25682\times 10^{6}$ Root an. cond. $6.51777$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 6·5-s − 84·11-s + 112·19-s + 146·25-s − 636·29-s − 104·31-s + 816·41-s + 616·49-s + 504·55-s + 372·59-s + 680·61-s − 72·71-s + 760·79-s + 2.23e3·89-s − 672·95-s + 2.24e3·101-s + 1.32e3·109-s − 176·121-s − 2.28e3·125-s + 127-s + 131-s + 137-s + 139-s + 3.81e3·145-s + 149-s + 151-s + 624·155-s + ⋯
 L(s)  = 1 − 0.536·5-s − 2.30·11-s + 1.35·19-s + 1.16·25-s − 4.07·29-s − 0.602·31-s + 3.10·41-s + 1.79·49-s + 1.23·55-s + 0.820·59-s + 1.42·61-s − 0.120·71-s + 1.08·79-s + 2.65·89-s − 0.725·95-s + 2.21·101-s + 1.16·109-s − 0.132·121-s − 1.63·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 2.18·145-s + 0.000549·149-s + 0.000538·151-s + 0.323·155-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{8} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$3.25682\times 10^{6}$$ Root analytic conductor: $$6.51777$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

Particular Values

 $$L(2)$$ $$\approx$$ $$4.538274207$$ $$L(\frac12)$$ $$\approx$$ $$4.538274207$$ $$L(\frac{5}{2})$$ 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 $$1$$
3 $$1$$
5$C_2^2$ $$1 + 6 T - 22 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}$$
good7$D_4\times C_2$ $$1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8}$$
11$D_{4}$ $$( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8}$$
17$D_4\times C_2$ $$1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 - 408 T + 177982 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8}$$
47$D_4\times C_2$ $$1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8}$$
59$D_{4}$ $$( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 36 T + 384046 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8}$$
79$D_{4}$ $$( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$