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 − 4·13-s + 4·17-s + 4·19-s − 10·23-s − 4·25-s − 4·29-s − 8·31-s − 8·35-s − 8·37-s + 2·41-s − 4·43-s − 8·47-s + 10·49-s + 16·53-s − 4·55-s − 24·59-s − 8·61-s − 8·65-s + 4·67-s − 10·71-s + 4·73-s + 8·77-s + 4·79-s − 20·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.970·17-s + 0.917·19-s − 2.08·23-s − 4/5·25-s − 0.742·29-s − 1.43·31-s − 1.35·35-s − 1.31·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 10/7·49-s + 2.19·53-s − 0.539·55-s − 3.12·59-s − 1.02·61-s − 0.992·65-s + 0.488·67-s − 1.18·71-s + 0.468·73-s + 0.911·77-s + 0.450·79-s − 2.19·83-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$C_2 \wr S_4$ \( 1 - 2 T + 8 T^{2} - 4 p T^{3} + 57 T^{4} - 4 p^{2} T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 2 T + 12 T^{2} - 48 T^{3} - 51 T^{4} - 48 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} - 20 T^{3} - 58 T^{4} - 20 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 36 T^{2} - 124 T^{3} + 694 T^{4} - 124 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 26 T^{2} + 40 T^{3} + 3 T^{4} + 40 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 10 T + 96 T^{2} + 576 T^{3} + 3385 T^{4} + 576 p T^{5} + 96 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 76 T^{2} + 172 T^{3} + 2694 T^{4} + 172 p T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8 T + 94 T^{2} + 392 T^{3} + 3303 T^{4} + 392 p T^{5} + 94 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 8 T + 62 T^{2} - 40 T^{3} - 121 T^{4} - 40 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2 T + 108 T^{2} - 4 T^{3} + 5237 T^{4} - 4 p T^{5} + 108 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 68 T^{2} - 220 T^{3} + 1014 T^{4} - 220 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 8 T + 124 T^{2} + 1016 T^{3} + 7926 T^{4} + 1016 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 148 T^{2} - 1136 T^{3} + 8534 T^{4} - 1136 p T^{5} + 148 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 24 T + 348 T^{2} + 3464 T^{3} + 29158 T^{4} + 3464 p T^{5} + 348 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 116 T^{2} + 824 T^{3} + 7478 T^{4} + 824 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 4 T + 100 T^{2} - 692 T^{3} + 4454 T^{4} - 692 p T^{5} + 100 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 10 T + 192 T^{2} + 1424 T^{3} + 19193 T^{4} + 1424 p T^{5} + 192 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 4 T + 44 T^{2} - 268 T^{3} + 10310 T^{4} - 268 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 404 T^{3} + 11814 T^{4} - 404 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 20 T + 436 T^{2} + 4932 T^{3} + 57734 T^{4} + 4932 p T^{5} + 436 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 26 T + 548 T^{2} - 7140 T^{3} + 80541 T^{4} - 7140 p T^{5} + 548 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 8 T + 196 T^{2} - 1688 T^{3} + 26118 T^{4} - 1688 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.17502573937938522782221174557, −5.60458293298297708253200440646, −5.60235096044450139553675153127, −5.60200748333670418485119132532, −5.57816072287223510890159894637, −5.05980781865816419816062421611, −5.02775255637902422328242432822, −4.77497781051528805131344695686, −4.75901140128448480615555119025, −4.13216308107884487842197387466, −3.99362171243244862418648001093, −3.91040217889449536433622994024, −3.86844915913480228476422684241, −3.48642541133279725901311015373, −3.25307792362347212445386809684, −3.21031589713688628771519693998, −2.91342307248750192133693783016, −2.59160637890866143515212707208, −2.30952654347509147752282504039, −2.30681072293080591246846384673, −2.22154332553312139924312248008, −1.58397742919684242623867525758, −1.48912965331237696264059651564, −1.27885108379442852892982304539, −1.10085068011544784272414270342, 0, 0, 0, 0, 1.10085068011544784272414270342, 1.27885108379442852892982304539, 1.48912965331237696264059651564, 1.58397742919684242623867525758, 2.22154332553312139924312248008, 2.30681072293080591246846384673, 2.30952654347509147752282504039, 2.59160637890866143515212707208, 2.91342307248750192133693783016, 3.21031589713688628771519693998, 3.25307792362347212445386809684, 3.48642541133279725901311015373, 3.86844915913480228476422684241, 3.91040217889449536433622994024, 3.99362171243244862418648001093, 4.13216308107884487842197387466, 4.75901140128448480615555119025, 4.77497781051528805131344695686, 5.02775255637902422328242432822, 5.05980781865816419816062421611, 5.57816072287223510890159894637, 5.60200748333670418485119132532, 5.60235096044450139553675153127, 5.60458293298297708253200440646, 6.17502573937938522782221174557

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