Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{8} \cdot 7^{8} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 2·8-s − 4·16-s − 8·29-s + 2·32-s + 16·58-s + 3·64-s − 8·116-s − 2·121-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·8-s − 4·16-s − 8·29-s + 2·32-s + 16·58-s + 3·64-s − 8·116-s − 2·121-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.1154271626\)
\(L(\frac12)\)  \(\approx\)  \(0.1154271626\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + T + T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^3$ \( 1 - T^{4} + T^{8} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
29$C_1$ \( ( 1 + T )^{8} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^3$ \( 1 - T^{4} + T^{8} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2^3$ \( 1 - T^{4} + T^{8} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2^3$ \( 1 - T^{4} + T^{8} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.10366130523002705997653649497, −7.02815258369489530782367059805, −6.35973050890619636035788423990, −6.15735368025110331681855042333, −6.11190708894644019578999840838, −5.65910093255353624535780162829, −5.53641733325952376367023765449, −5.34344098036039832425231811150, −5.29401768510124845186256481067, −4.83802481538778707204668618847, −4.82107656251439584939117241219, −4.19436994871180207183784261692, −4.12641046610760009615627387946, −3.93631867236240465507806718378, −3.86118595625707887904445289356, −3.50391329763465637679242450302, −3.26916145146759369928842089433, −2.93786377108900925863387528571, −2.44210338167708534506680586034, −2.04660180186693198617227803882, −1.91104522520550825427703019289, −1.71010602305280053563800764311, −1.59659538639705878876861769756, −0.947307761203043917555041474569, −0.31003323001282393933179788417, 0.31003323001282393933179788417, 0.947307761203043917555041474569, 1.59659538639705878876861769756, 1.71010602305280053563800764311, 1.91104522520550825427703019289, 2.04660180186693198617227803882, 2.44210338167708534506680586034, 2.93786377108900925863387528571, 3.26916145146759369928842089433, 3.50391329763465637679242450302, 3.86118595625707887904445289356, 3.93631867236240465507806718378, 4.12641046610760009615627387946, 4.19436994871180207183784261692, 4.82107656251439584939117241219, 4.83802481538778707204668618847, 5.29401768510124845186256481067, 5.34344098036039832425231811150, 5.53641733325952376367023765449, 5.65910093255353624535780162829, 6.11190708894644019578999840838, 6.15735368025110331681855042333, 6.35973050890619636035788423990, 7.02815258369489530782367059805, 7.10366130523002705997653649497

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