Properties

Degree 8
Conductor $ 3^{8} \cdot 7^{4} \cdot 11^{8} $
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  − 4·4-s + 4·7-s − 8·13-s + 4·16-s + 8·19-s − 6·25-s − 16·28-s − 8·31-s − 16·37-s − 4·43-s + 10·49-s + 32·52-s − 8·61-s + 16·64-s + 4·67-s − 48·73-s − 32·76-s − 24·79-s − 32·91-s + 16·97-s + 24·100-s − 8·103-s + 36·109-s + 16·112-s + 32·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2·4-s + 1.51·7-s − 2.21·13-s + 16-s + 1.83·19-s − 6/5·25-s − 3.02·28-s − 1.43·31-s − 2.63·37-s − 0.609·43-s + 10/7·49-s + 4.43·52-s − 1.02·61-s + 2·64-s + 0.488·67-s − 5.61·73-s − 3.67·76-s − 2.70·79-s − 3.35·91-s + 1.62·97-s + 12/5·100-s − 0.788·103-s + 3.44·109-s + 1.51·112-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 38 T^{2} + 715 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 32 T^{2} + 1090 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 56 T^{2} + 2242 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 + 158 T^{2} + 10435 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 28 T^{2} + 2230 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 158 T^{2} + 12307 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 4 T + 112 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 24 T + 276 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 174 T^{2} + 19331 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 198 T^{2} + 23627 T^{4} + 198 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.74061893542273244371402191894, −5.60404667570901566916544759158, −5.44149297265585979603581027937, −5.24291714489138113801193643407, −5.12449460439855861930337251663, −4.79657017080087931679147373510, −4.76632000089767777116205523627, −4.74262442628845806878341727539, −4.67794111145984956255797310069, −4.12274533110793621863411326408, −4.07535672915146882861037143408, −3.90800592944588166077255711569, −3.88392064425019373726242662572, −3.41035538409785950469209515587, −3.21948187128778142334188504910, −3.09412214047482753285799214724, −2.85768337670701013958104659881, −2.64797754110777421533853171006, −2.17182781535764302369022019343, −2.11718184544033026087738332169, −1.93086024471278529976048115838, −1.71042786823890710604135182518, −1.22197095139936685855159162192, −1.19189014388242068210239139133, −0.994143114623227906791068822615, 0, 0, 0, 0, 0.994143114623227906791068822615, 1.19189014388242068210239139133, 1.22197095139936685855159162192, 1.71042786823890710604135182518, 1.93086024471278529976048115838, 2.11718184544033026087738332169, 2.17182781535764302369022019343, 2.64797754110777421533853171006, 2.85768337670701013958104659881, 3.09412214047482753285799214724, 3.21948187128778142334188504910, 3.41035538409785950469209515587, 3.88392064425019373726242662572, 3.90800592944588166077255711569, 4.07535672915146882861037143408, 4.12274533110793621863411326408, 4.67794111145984956255797310069, 4.74262442628845806878341727539, 4.76632000089767777116205523627, 4.79657017080087931679147373510, 5.12449460439855861930337251663, 5.24291714489138113801193643407, 5.44149297265585979603581027937, 5.60404667570901566916544759158, 5.74061893542273244371402191894

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