Properties

Degree 8
Conductor $ 2^{16} \cdot 3^{8} \cdot 7^{4} $
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  − 10·7-s − 8·19-s + 5·25-s + 2·31-s − 8·37-s + 61·49-s + 36·61-s + 24·67-s − 24·73-s − 42·79-s − 16·103-s + 8·109-s − 7·121-s + 127-s + 131-s + 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s − 50·175-s + 179-s + ⋯
L(s)  = 1  − 3.77·7-s − 1.83·19-s + 25-s + 0.359·31-s − 1.31·37-s + 61/7·49-s + 4.60·61-s + 2.93·67-s − 2.80·73-s − 4.72·79-s − 1.57·103-s + 0.766·109-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + 6.93·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s − 3.77·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{16} \cdot 3^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.4870117246\)
\(L(\frac12)\)  \(\approx\)  \(0.4870117246\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good5$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 + 7 T^{2} - 72 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 14 T^{2} - 333 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 86 T^{2} + 5187 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 61 T^{2} + 912 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 73 T^{2} + 1848 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 18 T + 169 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 12 T + 115 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 118 T^{2} + 6003 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \)
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

−7.02422873041533328337385869638, −6.91234623730101930454115728588, −6.70728574643668517469699834824, −6.69079304558155988333959294635, −6.17443967569189105077382318803, −6.04379317209646630388971998786, −5.92573974734719527963069184734, −5.56849098980841387810046279967, −5.48139927121624473964138070826, −5.23117026569129646844917322087, −4.75388060518277368634393576174, −4.50561423919173909107225031819, −4.12452520169172423709139938571, −4.04772475234453435001718670517, −3.77767649679035371814330277991, −3.57417402299498779082517245199, −3.16055741595749354742153670064, −3.09178621235459058507616784607, −2.70833146071826184818772484193, −2.46177971153893970063453403682, −2.31621911310079668397047597300, −1.80282734843396788797130724223, −1.20503153995458890935588596464, −0.66661357118589378072445818320, −0.23574733024820085158381789164, 0.23574733024820085158381789164, 0.66661357118589378072445818320, 1.20503153995458890935588596464, 1.80282734843396788797130724223, 2.31621911310079668397047597300, 2.46177971153893970063453403682, 2.70833146071826184818772484193, 3.09178621235459058507616784607, 3.16055741595749354742153670064, 3.57417402299498779082517245199, 3.77767649679035371814330277991, 4.04772475234453435001718670517, 4.12452520169172423709139938571, 4.50561423919173909107225031819, 4.75388060518277368634393576174, 5.23117026569129646844917322087, 5.48139927121624473964138070826, 5.56849098980841387810046279967, 5.92573974734719527963069184734, 6.04379317209646630388971998786, 6.17443967569189105077382318803, 6.69079304558155988333959294635, 6.70728574643668517469699834824, 6.91234623730101930454115728588, 7.02422873041533328337385869638

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