Properties

Degree 8
Conductor $ 3^{4} \cdot 5^{4} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 2·5-s + 9-s + 12·11-s + 4·16-s − 12·19-s + 6·20-s + 5·25-s + 8·29-s + 20·31-s − 3·36-s + 8·41-s − 36·44-s − 2·45-s − 24·55-s − 16·59-s + 4·61-s − 9·64-s + 40·71-s + 36·76-s + 8·79-s − 8·80-s + 12·89-s + 24·95-s + 12·99-s − 15·100-s + 12·101-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.894·5-s + 1/3·9-s + 3.61·11-s + 16-s − 2.75·19-s + 1.34·20-s + 25-s + 1.48·29-s + 3.59·31-s − 1/2·36-s + 1.24·41-s − 5.42·44-s − 0.298·45-s − 3.23·55-s − 2.08·59-s + 0.512·61-s − 9/8·64-s + 4.74·71-s + 4.12·76-s + 0.900·79-s − 0.894·80-s + 1.27·89-s + 2.46·95-s + 1.20·99-s − 3/2·100-s + 1.19·101-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{735} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.37455\)
\(L(\frac12)\)  \(\approx\)  \(2.37455\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.75201671397994092120938549732, −6.90940117126105408411304863446, −6.86525569021167544476904604132, −6.84319662427915933036165948530, −6.52747827430576749522381841931, −6.28791704207243977077302512699, −6.19157767177446735150333500794, −6.09904079893725132173085593822, −5.64709156520941416091941830062, −5.02023216424947310413203488916, −4.83771595041845853617582873269, −4.66792903035204144243931585552, −4.49751735650570901220318794908, −4.46708317145231609878473002705, −3.94422141290013558236588923937, −3.79627073515957656390590813272, −3.71132493965297492633003591314, −3.52039805600232131296260805486, −2.78730957972345392662946025363, −2.58839088254207221525586711313, −2.30972295193615412235595740045, −1.60509465348059248411105966523, −1.33516815787608534265648526305, −0.78547004674032127317424196648, −0.65350587762320102125702073930, 0.65350587762320102125702073930, 0.78547004674032127317424196648, 1.33516815787608534265648526305, 1.60509465348059248411105966523, 2.30972295193615412235595740045, 2.58839088254207221525586711313, 2.78730957972345392662946025363, 3.52039805600232131296260805486, 3.71132493965297492633003591314, 3.79627073515957656390590813272, 3.94422141290013558236588923937, 4.46708317145231609878473002705, 4.49751735650570901220318794908, 4.66792903035204144243931585552, 4.83771595041845853617582873269, 5.02023216424947310413203488916, 5.64709156520941416091941830062, 6.09904079893725132173085593822, 6.19157767177446735150333500794, 6.28791704207243977077302512699, 6.52747827430576749522381841931, 6.84319662427915933036165948530, 6.86525569021167544476904604132, 6.90940117126105408411304863446, 7.75201671397994092120938549732

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