Properties

Degree 8
Conductor $ 2^{24} \cdot 3^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·5-s + 4·7-s + 6·9-s + 4·11-s − 12·13-s − 16·15-s + 4·19-s + 16·21-s + 8·25-s − 4·27-s − 12·29-s + 16·33-s − 16·35-s − 4·37-s − 48·39-s − 24·41-s + 4·43-s − 24·45-s + 10·49-s − 12·53-s − 16·55-s + 16·57-s + 12·59-s + 4·61-s + 24·63-s + 48·65-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 1.51·7-s + 2·9-s + 1.20·11-s − 3.32·13-s − 4.13·15-s + 0.917·19-s + 3.49·21-s + 8/5·25-s − 0.769·27-s − 2.22·29-s + 2.78·33-s − 2.70·35-s − 0.657·37-s − 7.68·39-s − 3.74·41-s + 0.609·43-s − 3.57·45-s + 10/7·49-s − 1.64·53-s − 2.15·55-s + 2.11·57-s + 1.56·59-s + 0.512·61-s + 3.02·63-s + 5.95·65-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{24} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1344} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{24} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.248486461$
$L(\frac12)$  $\approx$  $2.248486461$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 324 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 68 T^{3} + 574 T^{4} - 68 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
31$C_4\times C_2$ \( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_4$ \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 116 T^{3} + 1486 T^{4} - 116 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 660 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 82 T^{2} + p^{2} T^{4} ) \)
61$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 236 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 148 T^{2} + 14086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 444 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
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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

−7.05959415972599663549161393469, −6.89332091096994920880624372671, −6.80822121244550580333894949840, −6.20295677965717549238787329398, −6.08437173250393318372872227371, −5.44777015189044405786635025240, −5.40913998672384553376734622346, −5.17587520473510499240657918423, −5.06517143850924504364643675346, −4.95860008739282610995746630427, −4.47637591802946149056106556558, −4.16111010945247825257619121064, −3.98119724810994481960812186008, −3.95588747053071547790312406590, −3.72636817996185279744515505712, −3.14075889838702447559997436709, −3.10587736286391075381991735721, −3.08062174528199561501755240421, −2.63467017174363341050064987706, −2.08040230191164102626579712332, −2.05195571499784628574937193673, −1.76359661864946125204266022517, −1.65867709854020004461908263495, −0.77653286278372803848867569460, −0.27381194923424441313464692011, 0.27381194923424441313464692011, 0.77653286278372803848867569460, 1.65867709854020004461908263495, 1.76359661864946125204266022517, 2.05195571499784628574937193673, 2.08040230191164102626579712332, 2.63467017174363341050064987706, 3.08062174528199561501755240421, 3.10587736286391075381991735721, 3.14075889838702447559997436709, 3.72636817996185279744515505712, 3.95588747053071547790312406590, 3.98119724810994481960812186008, 4.16111010945247825257619121064, 4.47637591802946149056106556558, 4.95860008739282610995746630427, 5.06517143850924504364643675346, 5.17587520473510499240657918423, 5.40913998672384553376734622346, 5.44777015189044405786635025240, 6.08437173250393318372872227371, 6.20295677965717549238787329398, 6.80822121244550580333894949840, 6.89332091096994920880624372671, 7.05959415972599663549161393469

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