Properties

Degree 8
Conductor $ 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  − 2·2-s − 4-s − 4·5-s − 10·7-s + 4·8-s + 8·10-s − 2·11-s − 8·13-s + 20·14-s − 8·17-s − 2·19-s + 4·20-s + 4·22-s − 14·23-s + 5·25-s + 16·26-s + 10·28-s − 12·31-s − 2·32-s + 16·34-s + 40·35-s − 12·37-s + 4·38-s − 16·40-s − 6·43-s + 2·44-s + 28·46-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 1.78·5-s − 3.77·7-s + 1.41·8-s + 2.52·10-s − 0.603·11-s − 2.21·13-s + 5.34·14-s − 1.94·17-s − 0.458·19-s + 0.894·20-s + 0.852·22-s − 2.91·23-s + 25-s + 3.13·26-s + 1.88·28-s − 2.15·31-s − 0.353·32-s + 2.74·34-s + 6.76·35-s − 1.97·37-s + 0.648·38-s − 2.52·40-s − 0.914·43-s + 0.301·44-s + 4.12·46-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{315} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(8,\ 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 4 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 226 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 672 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$D_4\times C_2$ \( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 108 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 1224 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 600 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 140 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$$\times$$C_2^2$ \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} ) \)
97$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−9.152281283034785604741389051049, −8.561105879653770955758932865974, −8.458790976017573506282010299547, −8.424666384059848008544122160341, −8.317863359401090406320972685449, −7.57291525579443469611522539290, −7.40934533161736134437569534928, −7.38445168254937815534997134423, −7.14077475868531075746984718603, −6.66949696420315285969735052589, −6.65651615962366083596594509435, −6.20659110733079697382415269340, −6.01342747302789142007375845135, −5.70231369772050669472079267648, −5.27088811747544515284029233239, −4.98297382150262257174462376505, −4.56550392511836304380125359587, −4.16652355826738446727253117151, −4.05757220432945931809831945093, −3.77232174949928091027752476008, −3.46410635088639982306731617573, −3.00458943520830858995083971518, −2.89786278698498754613310790277, −2.15281136732298662894809818789, −2.01241289316343666063431252314, 0, 0, 0, 0, 2.01241289316343666063431252314, 2.15281136732298662894809818789, 2.89786278698498754613310790277, 3.00458943520830858995083971518, 3.46410635088639982306731617573, 3.77232174949928091027752476008, 4.05757220432945931809831945093, 4.16652355826738446727253117151, 4.56550392511836304380125359587, 4.98297382150262257174462376505, 5.27088811747544515284029233239, 5.70231369772050669472079267648, 6.01342747302789142007375845135, 6.20659110733079697382415269340, 6.65651615962366083596594509435, 6.66949696420315285969735052589, 7.14077475868531075746984718603, 7.38445168254937815534997134423, 7.40934533161736134437569534928, 7.57291525579443469611522539290, 8.317863359401090406320972685449, 8.424666384059848008544122160341, 8.458790976017573506282010299547, 8.561105879653770955758932865974, 9.152281283034785604741389051049

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