Properties

Degree 8
Conductor $ 3^{4} \cdot 23^{4} \cdot 29^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 9-s − 3·16-s − 2·18-s + 4·25-s + 2·31-s + 4·32-s + 2·36-s + 2·41-s − 2·47-s + 4·49-s − 8·50-s − 4·62-s − 2·64-s + 2·73-s − 4·82-s + 4·94-s − 8·98-s + 8·100-s − 4·101-s + 4·124-s + 127-s − 2·128-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s + 9-s − 3·16-s − 2·18-s + 4·25-s + 2·31-s + 4·32-s + 2·36-s + 2·41-s − 2·47-s + 4·49-s − 8·50-s − 4·62-s − 2·64-s + 2·73-s − 4·82-s + 4·94-s − 8·98-s + 8·100-s − 4·101-s + 4·124-s + 127-s − 2·128-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 23^{4} \cdot 29^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 3^{4} \cdot 23^{4} \cdot 29^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.7283346700$
$L(\frac12)$  $\approx$  $0.7283346700$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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

−6.74931738114314809733842765572, −6.70254386874767697018534394739, −6.62678357677016197280100112272, −6.16787685037953061378513476204, −6.02092574292142631439586123884, −5.69867129524085286195481193168, −5.45284178150281076795511102209, −5.22513946429756488004515141268, −4.93061985556174955312500072481, −4.71053927755874912365606573989, −4.63429578908161776856387020092, −4.35922411813746182554232787230, −4.15565750339888851861985690707, −4.08342838137541537382530050618, −3.46677617296152623034344502172, −3.39657362086873662715205391487, −3.06314956301727082962698382275, −2.61322163691978724226645359645, −2.41534826216741107223804090846, −2.38730044178192699650682811334, −2.05922790361499140130650471231, −1.37099445475855887632262427722, −1.29856665633053463670349467816, −0.915726320461567854752598713572, −0.915586976546599552619892365333, 0.915586976546599552619892365333, 0.915726320461567854752598713572, 1.29856665633053463670349467816, 1.37099445475855887632262427722, 2.05922790361499140130650471231, 2.38730044178192699650682811334, 2.41534826216741107223804090846, 2.61322163691978724226645359645, 3.06314956301727082962698382275, 3.39657362086873662715205391487, 3.46677617296152623034344502172, 4.08342838137541537382530050618, 4.15565750339888851861985690707, 4.35922411813746182554232787230, 4.63429578908161776856387020092, 4.71053927755874912365606573989, 4.93061985556174955312500072481, 5.22513946429756488004515141268, 5.45284178150281076795511102209, 5.69867129524085286195481193168, 6.02092574292142631439586123884, 6.16787685037953061378513476204, 6.62678357677016197280100112272, 6.70254386874767697018534394739, 6.74931738114314809733842765572

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