Properties

Degree 8
Conductor $ 2^{24} \cdot 3^{4} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 4

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·5-s + 10·9-s − 4·11-s − 8·13-s + 16·15-s + 4·17-s − 8·19-s + 12·23-s + 4·25-s − 20·27-s − 8·31-s + 16·33-s + 32·39-s + 20·41-s + 8·43-s − 40·45-s − 16·47-s − 16·51-s + 16·55-s + 32·57-s − 16·61-s + 32·65-s + 8·67-s − 48·69-s + 12·71-s + 8·73-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s + 10/3·9-s − 1.20·11-s − 2.21·13-s + 4.13·15-s + 0.970·17-s − 1.83·19-s + 2.50·23-s + 4/5·25-s − 3.84·27-s − 1.43·31-s + 2.78·33-s + 5.12·39-s + 3.12·41-s + 1.21·43-s − 5.96·45-s − 2.33·47-s − 2.24·51-s + 2.15·55-s + 4.23·57-s − 2.04·61-s + 3.96·65-s + 0.977·67-s − 5.77·69-s + 1.42·71-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{24} \cdot 3^{4} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(4\)
Selberg data  =  \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 98 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 24 T^{2} + 84 T^{3} + 302 T^{4} + 84 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 48 T^{2} + 232 T^{3} + 978 T^{4} + 232 p T^{5} + 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 44 T^{2} - 84 T^{3} + 818 T^{4} - 84 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 68 T^{2} + 392 T^{3} + 1926 T^{4} + 392 p T^{5} + 68 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 120 T^{2} - 780 T^{3} + 4350 T^{4} - 780 p T^{5} + 120 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 60 T^{2} - 128 T^{3} + 1862 T^{4} - 128 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 84 T^{2} + 616 T^{3} + 3222 T^{4} + 616 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 220 T^{2} - 1540 T^{3} + 9874 T^{4} - 1540 p T^{5} + 220 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 648 T^{3} + 6710 T^{4} - 648 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 164 T^{2} + 1232 T^{3} + 170 p T^{4} + 1232 p T^{5} + 164 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 140 T^{2} + 128 T^{3} + 9494 T^{4} + 128 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 84 T^{2} - 960 T^{3} + 1350 T^{4} - 960 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 288 T^{2} + 2768 T^{3} + 27282 T^{4} + 2768 p T^{5} + 288 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} + 56 T^{3} + 3286 T^{4} + 56 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 248 T^{2} - 2380 T^{3} + 25406 T^{4} - 2380 p T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 256 T^{2} - 1608 T^{3} + 27170 T^{4} - 1608 p T^{5} + 256 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 284 T^{2} - 2768 T^{3} + 31366 T^{4} - 2768 p T^{5} + 284 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 204 T^{2} - 256 T^{3} + 21878 T^{4} - 256 p T^{5} + 204 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 524 T^{2} - 76 p T^{3} + 72658 T^{4} - 76 p^{2} T^{5} + 524 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 896 T^{2} - 13608 T^{3} + 153858 T^{4} - 13608 p T^{5} + 896 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.86176884912625710050504860392, −5.36586937994949186008391280612, −5.30270751690245038332371245862, −5.09393083762035683684041921622, −5.09112492114489264301380767606, −4.80334406988242374780926916274, −4.80232429548997645181576808257, −4.63353628871924754458417855234, −4.58265447813848175119506277788, −3.95490298626924780592256400373, −3.94288539409999995425598439883, −3.90162253843110061224610998685, −3.73196127845058341475018403898, −3.40366494551501055121477188071, −3.20429494256637114063498794092, −2.88540342648179512321213156606, −2.84035376146860658487498264605, −2.24761283559428606605929140686, −2.18066229712472413086567410327, −2.17391400848036015800676030013, −2.16274980201029654054923278701, −1.24477460300249478762871513902, −1.13836171604517499477148773315, −0.955707403670205989187915268043, −0.871428562636313152813093370774, 0, 0, 0, 0, 0.871428562636313152813093370774, 0.955707403670205989187915268043, 1.13836171604517499477148773315, 1.24477460300249478762871513902, 2.16274980201029654054923278701, 2.17391400848036015800676030013, 2.18066229712472413086567410327, 2.24761283559428606605929140686, 2.84035376146860658487498264605, 2.88540342648179512321213156606, 3.20429494256637114063498794092, 3.40366494551501055121477188071, 3.73196127845058341475018403898, 3.90162253843110061224610998685, 3.94288539409999995425598439883, 3.95490298626924780592256400373, 4.58265447813848175119506277788, 4.63353628871924754458417855234, 4.80232429548997645181576808257, 4.80334406988242374780926916274, 5.09112492114489264301380767606, 5.09393083762035683684041921622, 5.30270751690245038332371245862, 5.36586937994949186008391280612, 5.86176884912625710050504860392

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