Properties

Degree 8
Conductor $ 2^{20} \cdot 3^{12} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 4

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s + 2·11-s − 8·17-s + 4·19-s + 2·23-s − 4·25-s − 8·29-s − 4·31-s + 8·35-s − 4·37-s − 2·41-s + 8·43-s + 12·47-s + 10·49-s − 16·53-s − 4·55-s + 12·59-s − 8·61-s − 8·67-s − 18·71-s + 8·73-s − 8·77-s + 16·85-s − 18·89-s − 8·95-s + 8·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 0.603·11-s − 1.94·17-s + 0.917·19-s + 0.417·23-s − 4/5·25-s − 1.48·29-s − 0.718·31-s + 1.35·35-s − 0.657·37-s − 0.312·41-s + 1.21·43-s + 1.75·47-s + 10/7·49-s − 2.19·53-s − 0.539·55-s + 1.56·59-s − 1.02·61-s − 0.977·67-s − 2.13·71-s + 0.936·73-s − 0.911·77-s + 1.73·85-s − 1.90·89-s − 0.820·95-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 2 T + 8 T^{2} + 24 T^{3} + 33 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 T + 20 T^{2} + 20 T^{3} + 125 T^{4} + 20 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 8 T + 44 T^{2} + 184 T^{3} + 854 T^{4} + 184 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 34 T^{2} - 72 T^{3} + 699 T^{4} - 72 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} - 4 T^{3} + 761 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 8 T + 80 T^{2} + 296 T^{3} + 2206 T^{4} + 296 p T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 4 T + 70 T^{2} + 448 T^{3} + 2407 T^{4} + 448 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 344 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 2 T + 92 T^{2} + 80 T^{3} + 4093 T^{4} + 80 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 8 T + 160 T^{2} - 952 T^{3} + 10046 T^{4} - 952 p T^{5} + 160 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 12 T + 200 T^{2} - 1484 T^{3} + 13950 T^{4} - 1484 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
59$C_2 \wr S_4$ \( 1 - 12 T + 248 T^{2} - 1916 T^{3} + 21870 T^{4} - 1916 p T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 440 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 8 T + 88 T^{2} + 8 T^{3} + 814 T^{4} + 8 p T^{5} + 88 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 18 T + 296 T^{2} + 2724 T^{3} + 27369 T^{4} + 2724 p T^{5} + 296 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 1032 p T^{5} + 136 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 160 T^{2} - 848 T^{3} + 11886 T^{4} - 848 p T^{5} + 160 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 40 T^{2} - 16 T^{3} + 13134 T^{4} - 16 p T^{5} - 40 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 18 T + 404 T^{2} + 4416 T^{3} + 54549 T^{4} + 4416 p T^{5} + 404 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 2200 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.19720813844065877036945559196, −5.66118250082932931228209481277, −5.65038130369142942981836975726, −5.59565222499160576465009108261, −5.53917169568997885875609847592, −4.94445792799587683605361185019, −4.92414424898903618862685177665, −4.82217021622557919878835899680, −4.45167745989121934425277728738, −4.11395320790515154638683138406, −4.01928868620050316680084386365, −3.98324755427311215372226591526, −3.92923182619190741137687358340, −3.55505141096117627634723793815, −3.30130433968494897717015539923, −3.16353609345142643925700885623, −3.05693674830878706285170586134, −2.50280131550290241217258821165, −2.45722446900485592197474343902, −2.31055093673384561781893488543, −2.24772974770745639588161047866, −1.54253869646833641274836028975, −1.36033027197864957309686515364, −1.22556150059834491286759426717, −1.05911064826559099409753438687, 0, 0, 0, 0, 1.05911064826559099409753438687, 1.22556150059834491286759426717, 1.36033027197864957309686515364, 1.54253869646833641274836028975, 2.24772974770745639588161047866, 2.31055093673384561781893488543, 2.45722446900485592197474343902, 2.50280131550290241217258821165, 3.05693674830878706285170586134, 3.16353609345142643925700885623, 3.30130433968494897717015539923, 3.55505141096117627634723793815, 3.92923182619190741137687358340, 3.98324755427311215372226591526, 4.01928868620050316680084386365, 4.11395320790515154638683138406, 4.45167745989121934425277728738, 4.82217021622557919878835899680, 4.92414424898903618862685177665, 4.94445792799587683605361185019, 5.53917169568997885875609847592, 5.59565222499160576465009108261, 5.65038130369142942981836975726, 5.66118250082932931228209481277, 6.19720813844065877036945559196

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