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

−6.00485720056768267521795581375, −5.84022284715977683623071890315, −5.65829777198799405776037601016, −5.64517033927695183228115278100, −5.21031478428585827598999763239, −4.82526015201858381392429162724, −4.82476686547887270950514035331, −4.72959347910585112068780688628, −4.70821623877111662651496583957, −4.29547302694165690376662993102, −4.15031596490727025965183821296, −4.14526111327215882874131261613, −3.92907537113808629967662038523, −3.45338333184067653435572420290, −3.27897664057588482363945842174, −3.16676449276260505445667389202, −2.90652485638411626486290873628, −2.60088430367830208176508588291, −2.33724498217187129317758827266, −2.31559528601382538606644926080, −1.98683734026407773391228681533, −1.66163234802565522992839509855, −1.52975403692924058925749713638, −1.26754755950454492710176783200, −1.06877555629821362810733625144, 0, 0, 0, 0, 1.06877555629821362810733625144, 1.26754755950454492710176783200, 1.52975403692924058925749713638, 1.66163234802565522992839509855, 1.98683734026407773391228681533, 2.31559528601382538606644926080, 2.33724498217187129317758827266, 2.60088430367830208176508588291, 2.90652485638411626486290873628, 3.16676449276260505445667389202, 3.27897664057588482363945842174, 3.45338333184067653435572420290, 3.92907537113808629967662038523, 4.14526111327215882874131261613, 4.15031596490727025965183821296, 4.29547302694165690376662993102, 4.70821623877111662651496583957, 4.72959347910585112068780688628, 4.82476686547887270950514035331, 4.82526015201858381392429162724, 5.21031478428585827598999763239, 5.64517033927695183228115278100, 5.65829777198799405776037601016, 5.84022284715977683623071890315, 6.00485720056768267521795581375

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