Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 12·16-s + 12·25-s + 48·37-s − 32·64-s − 48·100-s − 24·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 192·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s + 12/5·25-s + 7.89·37-s − 4·64-s − 4.79·100-s − 2.29·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.7·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.619622227\)
\(L(\frac12)\)  \(\approx\)  \(2.619622227\)
\(L(\frac{3}{2})\)   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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 120 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.44981964668487474632504286682, −6.32956978148163461379656360116, −6.31623916961709260011527901753, −5.87136029994254922647475097805, −5.72287064771039191265290200714, −5.56981111621362003000498615326, −5.38247457313729314879864001835, −4.87907470592395728979913472387, −4.75798195623369930991834351556, −4.73719201562354593869658620435, −4.57582245834994345713036511453, −4.12521246856687554544636401912, −4.02452422995861658736526581565, −3.85288770084505224453031272397, −3.76150245797625627700967242535, −2.99887407347835740493424906111, −2.98436293698379732212671002035, −2.74440393467039269855684966733, −2.66745170882070923274634382903, −2.29555633457514920523112979363, −1.68852368127596592519540870803, −1.17045453004939814686854578233, −1.09483860997592126830343195965, −0.78552115538202780571156344868, −0.42870500871592043268282414520, 0.42870500871592043268282414520, 0.78552115538202780571156344868, 1.09483860997592126830343195965, 1.17045453004939814686854578233, 1.68852368127596592519540870803, 2.29555633457514920523112979363, 2.66745170882070923274634382903, 2.74440393467039269855684966733, 2.98436293698379732212671002035, 2.99887407347835740493424906111, 3.76150245797625627700967242535, 3.85288770084505224453031272397, 4.02452422995861658736526581565, 4.12521246856687554544636401912, 4.57582245834994345713036511453, 4.73719201562354593869658620435, 4.75798195623369930991834351556, 4.87907470592395728979913472387, 5.38247457313729314879864001835, 5.56981111621362003000498615326, 5.72287064771039191265290200714, 5.87136029994254922647475097805, 6.31623916961709260011527901753, 6.32956978148163461379656360116, 6.44981964668487474632504286682

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