Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{4} \cdot 67^{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  + 4·9-s − 12·19-s + 20·37-s + 24·49-s − 28·67-s + 20·73-s + 7·81-s + 40·103-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s − 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 4/3·9-s − 2.75·19-s + 3.28·37-s + 24/7·49-s − 3.42·67-s + 2.34·73-s + 7/9·81-s + 3.94·103-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-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 + 1.23·169-s − 3.67·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{4} \cdot 67^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{804} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 3^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.57439$
$L(\frac12)$  $\approx$  $2.57439$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 89 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 173 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
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

−7.46985906217230104649218205236, −7.34822885380022822483018009780, −6.89916270789847921638707219963, −6.43960915658412676994465965770, −6.43563795276384791839368163809, −6.36067843233360608287864755165, −6.07826747486617672411190416660, −5.88143900753117517513427491639, −5.39442805229894832653822403570, −5.37945793692083760899248127196, −4.77892523546812327037685112636, −4.77324740785394044542911710850, −4.38563512973051706941597439941, −4.20014960372126109493567815418, −4.03755883383147708236504169715, −3.95467863585309783018298840209, −3.39686480807023915682682180802, −3.19376664674168012437709767066, −2.54219591982894105706580875697, −2.45590510625300637903520791201, −2.29009983163835698543836546844, −1.89843080662219645565068105752, −1.31851204504182609025724914495, −1.05232544094010160938320677403, −0.44631356795939100241948177450, 0.44631356795939100241948177450, 1.05232544094010160938320677403, 1.31851204504182609025724914495, 1.89843080662219645565068105752, 2.29009983163835698543836546844, 2.45590510625300637903520791201, 2.54219591982894105706580875697, 3.19376664674168012437709767066, 3.39686480807023915682682180802, 3.95467863585309783018298840209, 4.03755883383147708236504169715, 4.20014960372126109493567815418, 4.38563512973051706941597439941, 4.77324740785394044542911710850, 4.77892523546812327037685112636, 5.37945793692083760899248127196, 5.39442805229894832653822403570, 5.88143900753117517513427491639, 6.07826747486617672411190416660, 6.36067843233360608287864755165, 6.43563795276384791839368163809, 6.43960915658412676994465965770, 6.89916270789847921638707219963, 7.34822885380022822483018009780, 7.46985906217230104649218205236

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