Properties

 Label 8-5070e4-1.1-c1e4-0-2 Degree $8$ Conductor $6.607\times 10^{14}$ Sign $1$ Analytic cond. $2.68621\times 10^{6}$ Root an. cond. $6.36271$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 3·16-s + 16·17-s − 2·25-s − 20·27-s − 8·29-s − 20·36-s + 24·43-s − 12·48-s + 10·49-s − 64·51-s − 8·53-s − 16·61-s − 4·64-s − 32·68-s + 8·75-s + 40·79-s + 35·81-s + 32·87-s + 4·100-s + 8·101-s + 16·103-s − 24·107-s + 40·108-s + ⋯
 L(s)  = 1 − 2.30·3-s − 4-s + 10/3·9-s + 2.30·12-s + 3/4·16-s + 3.88·17-s − 2/5·25-s − 3.84·27-s − 1.48·29-s − 3.33·36-s + 3.65·43-s − 1.73·48-s + 10/7·49-s − 8.96·51-s − 1.09·53-s − 2.04·61-s − 1/2·64-s − 3.88·68-s + 0.923·75-s + 4.50·79-s + 35/9·81-s + 3.43·87-s + 2/5·100-s + 0.796·101-s + 1.57·103-s − 2.32·107-s + 3.84·108-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$2.68621\times 10^{6}$$ Root analytic conductor: $$6.36271$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{5070} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.299451173$$ $$L(\frac12)$$ $$\approx$$ $$2.299451173$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_1$ $$( 1 + T )^{4}$$
5$C_2$ $$( 1 + T^{2} )^{2}$$
13 $$1$$
good7$C_2^2$ $$( 1 - 5 T^{2} + p^{2} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
19$D_4\times C_2$ $$1 - 2 p T^{2} + 891 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + 34 T^{2} + p^{2} T^{4} )^{2}$$
29$D_{4}$ $$( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 50 T^{2} + 3171 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 - 66 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
47$D_4\times C_2$ $$1 - 146 T^{2} + 9315 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 84 T^{2} + 5654 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 116 T^{2} + 6534 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 98 T^{2} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 54 T^{2} + 14219 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 164 T^{2} + 23814 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$