Properties

Degree 8
Conductor $ 3^{4} \cdot 5^{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 − 2·4-s − 4·7-s + 6·9-s − 8·12-s + 16·13-s − 5·16-s − 16·21-s + 2·25-s − 4·27-s + 8·28-s − 12·36-s + 64·39-s − 20·48-s − 2·49-s − 32·52-s − 24·63-s + 20·64-s − 32·73-s + 8·75-s + 32·79-s − 37·81-s + 32·84-s − 64·91-s − 32·97-s − 4·100-s + 40·103-s + ⋯
L(s)  = 1  + 2.30·3-s − 4-s − 1.51·7-s + 2·9-s − 2.30·12-s + 4.43·13-s − 5/4·16-s − 3.49·21-s + 2/5·25-s − 0.769·27-s + 1.51·28-s − 2·36-s + 10.2·39-s − 2.88·48-s − 2/7·49-s − 4.43·52-s − 3.02·63-s + 5/2·64-s − 3.74·73-s + 0.923·75-s + 3.60·79-s − 4.11·81-s + 3.49·84-s − 6.70·91-s − 3.24·97-s − 2/5·100-s + 3.94·103-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.50124$
$L(\frac12)$  $\approx$  $1.50124$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
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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

−10.11856356347458545064561785431, −9.513682576110652253692021784581, −9.501650525825397968690254685471, −9.133970793684972294741362444975, −9.059678842927342555093971210870, −8.848326879158142793499529862373, −8.442539820864108846324136547569, −8.283893694847508889487503317621, −8.188854155736163205269556742399, −7.86847010231486027892400878543, −7.11828797467679815132894401824, −6.99425896532514950415498876883, −6.61545921427657028036280746754, −6.19409117106424406375678282124, −5.92651193369505600049189576146, −5.82147678121895778841274969685, −5.14419943791786367584315889687, −4.45085844427355010240983219569, −4.22850298330842497974047409690, −3.77425723124847594775547346659, −3.43890404297096500369201921575, −3.35833776691349154454458590662, −2.87845056192373894188141371571, −2.20572488300709862004873935729, −1.42578641308288919280803743557, 1.42578641308288919280803743557, 2.20572488300709862004873935729, 2.87845056192373894188141371571, 3.35833776691349154454458590662, 3.43890404297096500369201921575, 3.77425723124847594775547346659, 4.22850298330842497974047409690, 4.45085844427355010240983219569, 5.14419943791786367584315889687, 5.82147678121895778841274969685, 5.92651193369505600049189576146, 6.19409117106424406375678282124, 6.61545921427657028036280746754, 6.99425896532514950415498876883, 7.11828797467679815132894401824, 7.86847010231486027892400878543, 8.188854155736163205269556742399, 8.283893694847508889487503317621, 8.442539820864108846324136547569, 8.848326879158142793499529862373, 9.059678842927342555093971210870, 9.133970793684972294741362444975, 9.501650525825397968690254685471, 9.513682576110652253692021784581, 10.11856356347458545064561785431

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