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

−6.14260017171135465786573044022, −5.84915774576260205627800573054, −5.46832726969435100595412486583, −5.45544449305825601215247146264, −5.23190982466024415953290872661, −5.07951553987441292718820215810, −4.89339692215446714803052102984, −4.85335956505728573845502537659, −4.43271897365286880668708882343, −4.37647022448244239499533932173, −4.14623802162842325790009360883, −3.96372480359134478396832746046, −3.89900381959242159029832469977, −3.50147236565789978357316949051, −3.35389309763696306449875618557, −3.33152625836231481214124990423, −2.73770170299340144041187481799, −2.53438157053937381774033938180, −2.48048763121543458375746276454, −2.22526209014727458417452950301, −2.05885920089271034156037459578, −1.51536851296709074778771312522, −1.49212911354658975128834941193, −1.43097230627553570402284655104, −1.05574691377499523292259938640, 0, 0, 0, 0, 1.05574691377499523292259938640, 1.43097230627553570402284655104, 1.49212911354658975128834941193, 1.51536851296709074778771312522, 2.05885920089271034156037459578, 2.22526209014727458417452950301, 2.48048763121543458375746276454, 2.53438157053937381774033938180, 2.73770170299340144041187481799, 3.33152625836231481214124990423, 3.35389309763696306449875618557, 3.50147236565789978357316949051, 3.89900381959242159029832469977, 3.96372480359134478396832746046, 4.14623802162842325790009360883, 4.37647022448244239499533932173, 4.43271897365286880668708882343, 4.85335956505728573845502537659, 4.89339692215446714803052102984, 5.07951553987441292718820215810, 5.23190982466024415953290872661, 5.45544449305825601215247146264, 5.46832726969435100595412486583, 5.84915774576260205627800573054, 6.14260017171135465786573044022

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