# Properties

 Label 8-12e12-1.1-c0e4-0-5 Degree $8$ Conductor $8.916\times 10^{12}$ Sign $1$ Analytic cond. $0.553099$ Root an. cond. $0.928646$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s + 2·13-s + 3·25-s − 2·29-s + 2·41-s − 49-s + 2·61-s + 4·65-s − 2·97-s + 2·101-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 + 2·5-s + 2·13-s + 3·25-s − 2·29-s + 2·41-s − 49-s + 2·61-s + 4·65-s − 2·97-s + 2·101-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 3^{12}$$ Sign: $1$ Analytic conductor: $$0.553099$$ Root analytic conductor: $$0.928646$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.331593264$$ $$L(\frac12)$$ $$\approx$$ $$2.331593264$$ $$L(1)$$ 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 $$1$$
good5$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
7$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
11$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
13$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
17$C_2$ $$( 1 + T^{2} )^{4}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
23$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
29$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
31$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
37$C_2$ $$( 1 + T^{2} )^{4}$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
43$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
47$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
53$C_2$ $$( 1 + T^{2} )^{4}$$
59$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
67$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
71$C_2$ $$( 1 + T^{2} )^{4}$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
83$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
89$C_2$ $$( 1 + T^{2} )^{4}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−6.87934197690069529202837490502, −6.52232047004454338084035725251, −6.45671470629239178836423714981, −5.94774051497234796275808604794, −5.93230029871973798035470081817, −5.90081304946985317127610828765, −5.68389307123091056654112544851, −5.49776867271052208112105938889, −5.24248666134070464560400147146, −4.90457158059993175440515498001, −4.66259989987023571304349153822, −4.46405782510757234638656158648, −4.38707309781617242343030375863, −3.73866777472437973757336750750, −3.59078343140711757516828772445, −3.53293102406919621341779264794, −3.48679898519357348431707504018, −2.74950149137168911424797330324, −2.62627162503586019791877367352, −2.38665520872311839944329547963, −2.20187990991086008338375275662, −1.70005822714374503798060240629, −1.56310958133187847681320844723, −1.12131364256116508425898221565, −0.987392758379767511786723518841, 0.987392758379767511786723518841, 1.12131364256116508425898221565, 1.56310958133187847681320844723, 1.70005822714374503798060240629, 2.20187990991086008338375275662, 2.38665520872311839944329547963, 2.62627162503586019791877367352, 2.74950149137168911424797330324, 3.48679898519357348431707504018, 3.53293102406919621341779264794, 3.59078343140711757516828772445, 3.73866777472437973757336750750, 4.38707309781617242343030375863, 4.46405782510757234638656158648, 4.66259989987023571304349153822, 4.90457158059993175440515498001, 5.24248666134070464560400147146, 5.49776867271052208112105938889, 5.68389307123091056654112544851, 5.90081304946985317127610828765, 5.93230029871973798035470081817, 5.94774051497234796275808604794, 6.45671470629239178836423714981, 6.52232047004454338084035725251, 6.87934197690069529202837490502