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

Origins of factors

Downloads

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Normalization:  

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

Graph of the $Z$-function along the critical line