Properties

 Label 8-117e4-1.1-c3e4-0-4 Degree $8$ Conductor $187388721$ Sign $1$ Analytic cond. $2270.95$ Root an. cond. $2.62739$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 115·16-s + 1.80e3·43-s + 1.37e3·49-s + 3.39e3·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
 L(s)  = 1 − 1.79·16-s + 6.41·43-s + 4·49-s + 3.24·103-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$2270.95$$ Root analytic conductor: $$2.62739$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

Particular Values

 $$L(2)$$ $$\approx$$ $$3.366005925$$ $$L(\frac12)$$ $$\approx$$ $$3.366005925$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
13$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
good2$D_4\times C_2$ $$1 + 115 T^{4} + p^{12} T^{8}$$
5$D_4\times C_2$ $$1 + 3742 T^{4} + p^{12} T^{8}$$
7$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
11$D_4\times C_2$ $$1 - 1486370 T^{4} + p^{12} T^{8}$$
17$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
19$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
23$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
29$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
31$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
37$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
41$D_4\times C_2$ $$1 - 315317810 T^{4} + p^{12} T^{8}$$
43$C_2$ $$( 1 - 452 T + p^{3} T^{2} )^{4}$$
47$D_4\times C_2$ $$1 - 19480835090 T^{4} + p^{12} T^{8}$$
53$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
59$D_4\times C_2$ $$1 + 78746477470 T^{4} + p^{12} T^{8}$$
61$C_2^2$ $$( 1 - 438410 T^{2} + p^{6} T^{4} )^{2}$$
67$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
71$D_4\times C_2$ $$1 - 234490873970 T^{4} + p^{12} T^{8}$$
73$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
79$C_2^2$ $$( 1 + 811150 T^{2} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 633473875010 T^{4} + p^{12} T^{8}$$
89$D_4\times C_2$ $$1 - 926581329650 T^{4} + p^{12} T^{8}$$
97$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$