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Properties

Label 8-2312e4-1.1-c0e4-0-3
Degree $8$
Conductor $2.857\times 10^{13}$
Sign $1$
Analytic cond. $1.77247$
Root an. cond. $1.07416$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16-s + 8·67-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 241-s + 251-s + ⋯
L(s)  = 1  − 16-s + 8·67-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(1.77247\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421686980\)
\(L(\frac12)\) \(\approx\) \(1.421686980\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
17 \( 1 \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_4\times C_2$ \( 1 + T^{8} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$ \( ( 1 - T )^{8} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_4\times C_2$ \( 1 + T^{8} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_4\times C_2$ \( 1 + T^{8} \)
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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

−6.45342254332649190769180668322, −6.43433094806928459563688170687, −6.24645980435625160952427942308, −6.07559292753882842868518072876, −5.61462328686887477443063831466, −5.55420115214309402874914595729, −5.43560006535886436062470569220, −5.00542919460130982611186626472, −4.95445696055809479436942117576, −4.76431076193265845511270898639, −4.60894539258318054103948314037, −4.10785915539618027550561037447, −3.92104226276167143644047942918, −3.87443872781623881171309099983, −3.76389476188217843196402056899, −3.13538491372726399786320898136, −3.13336257552396451774035909162, −2.97839552388439962333696984792, −2.33974253115638221557235753819, −2.26494508970013943867371643918, −2.09267031624450672114732513965, −1.91442494301080118043109880529, −1.38493904020624041157621333140, −0.799591560594038470159497330029, −0.75754313721658315546386503779, 0.75754313721658315546386503779, 0.799591560594038470159497330029, 1.38493904020624041157621333140, 1.91442494301080118043109880529, 2.09267031624450672114732513965, 2.26494508970013943867371643918, 2.33974253115638221557235753819, 2.97839552388439962333696984792, 3.13336257552396451774035909162, 3.13538491372726399786320898136, 3.76389476188217843196402056899, 3.87443872781623881171309099983, 3.92104226276167143644047942918, 4.10785915539618027550561037447, 4.60894539258318054103948314037, 4.76431076193265845511270898639, 4.95445696055809479436942117576, 5.00542919460130982611186626472, 5.43560006535886436062470569220, 5.55420115214309402874914595729, 5.61462328686887477443063831466, 6.07559292753882842868518072876, 6.24645980435625160952427942308, 6.43433094806928459563688170687, 6.45342254332649190769180668322

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