Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{8} \cdot 5^{4} \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  − 2·4-s − 2·5-s − 10·7-s − 3·9-s − 6·11-s − 12·13-s + 3·16-s + 6·19-s + 4·20-s − 12·23-s + 25-s + 20·28-s − 12·29-s + 20·35-s + 6·36-s − 2·37-s + 18·41-s + 4·43-s + 12·44-s + 6·45-s − 36·47-s + 61·49-s + 24·52-s − 6·53-s + 12·55-s − 12·59-s + 30·63-s + ⋯
L(s)  = 1  − 4-s − 0.894·5-s − 3.77·7-s − 9-s − 1.80·11-s − 3.32·13-s + 3/4·16-s + 1.37·19-s + 0.894·20-s − 2.50·23-s + 1/5·25-s + 3.77·28-s − 2.22·29-s + 3.38·35-s + 36-s − 0.328·37-s + 2.81·41-s + 0.609·43-s + 1.80·44-s + 0.894·45-s − 5.25·47-s + 61/7·49-s + 3.32·52-s − 0.824·53-s + 1.61·55-s − 1.56·59-s + 3.77·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{630} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(4\)
Selberg data  =  \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good11$D_4\times C_2$ \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 12 T + 70 T^{2} + 264 T^{3} + 1059 T^{4} + 264 p T^{5} + 70 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 52 p T^{5} - 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 92 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
53$D_4\times C_2$ \( 1 + 6 T + 40 T^{2} + 168 T^{3} - 1389 T^{4} + 168 p T^{5} + 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 12 T - 31 T^{2} + 108 T^{3} + 11784 T^{4} + 108 p T^{5} - 31 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 12 T + 38 T^{2} - 864 T^{3} - 9501 T^{4} - 864 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 12 T + 110 T^{2} - 744 T^{3} - 909 T^{4} - 744 p T^{5} + 110 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.936163045557615542462880159866, −7.79517787228108252929192716538, −7.67797162887753511526762695191, −7.45580786820551258123825439594, −7.41877468704941256557466854515, −6.90127023460504338083326593194, −6.55035491725307089210655902052, −6.53037265933990513408761694785, −6.18569606829290761853384528277, −6.07754067367631429430677093575, −5.51329322145192892682867486972, −5.39415029064606927896043746479, −5.36051672880127327013806407837, −4.96785759406228997175136856123, −4.91080575512478723561876561556, −4.09414946486629084257041971596, −3.98810345368101453656271810666, −3.96578539162292035218827257301, −3.69831776505631669409499329688, −2.90615923394850078885018616289, −2.87972274484168293399399826024, −2.82510419271929226058787298081, −2.79502513973306058405091944143, −2.14615554663578379021417444230, −1.48673456284610144314278423914, 0, 0, 0, 0, 1.48673456284610144314278423914, 2.14615554663578379021417444230, 2.79502513973306058405091944143, 2.82510419271929226058787298081, 2.87972274484168293399399826024, 2.90615923394850078885018616289, 3.69831776505631669409499329688, 3.96578539162292035218827257301, 3.98810345368101453656271810666, 4.09414946486629084257041971596, 4.91080575512478723561876561556, 4.96785759406228997175136856123, 5.36051672880127327013806407837, 5.39415029064606927896043746479, 5.51329322145192892682867486972, 6.07754067367631429430677093575, 6.18569606829290761853384528277, 6.53037265933990513408761694785, 6.55035491725307089210655902052, 6.90127023460504338083326593194, 7.41877468704941256557466854515, 7.45580786820551258123825439594, 7.67797162887753511526762695191, 7.79517787228108252929192716538, 7.936163045557615542462880159866

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