Properties

Degree 8
Conductor $ 2^{12} \cdot 3^{4} \cdot 5^{8} $
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-s + 3-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯
L(s)  = 1  + 2-s + 3-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{12} \cdot 3^{4} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.9983621939$
$L(\frac12)$  $\approx$  $0.9983621939$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
good7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
31$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
59$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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

−7.84232125263057793666246361886, −7.66981501816775883330459380997, −7.43258593177356800160634817136, −7.14254561741020520523290292026, −6.97670638188966200650886100836, −6.86649025843642921970021235882, −6.16801099675516020279824416309, −6.02856331054450554983654699819, −6.01834080393432966584567233959, −5.89463443287140223083459360678, −5.69430914124502981247248948137, −5.07288703322210171428009826633, −4.75093527236746108051610999977, −4.73415586335395170227535938226, −4.54041723275158527535589520431, −4.46165948994787499370132581884, −3.67476107723092059868908245222, −3.43394796637693852198834991532, −3.10520709311688864591500395430, −3.03312472714709196783661963924, −2.70509528835814710740052894529, −2.62389902806177915964267520957, −2.15081326373838919591372847816, −1.82616607877136577685655652354, −0.890317674643063229284811383072, 0.890317674643063229284811383072, 1.82616607877136577685655652354, 2.15081326373838919591372847816, 2.62389902806177915964267520957, 2.70509528835814710740052894529, 3.03312472714709196783661963924, 3.10520709311688864591500395430, 3.43394796637693852198834991532, 3.67476107723092059868908245222, 4.46165948994787499370132581884, 4.54041723275158527535589520431, 4.73415586335395170227535938226, 4.75093527236746108051610999977, 5.07288703322210171428009826633, 5.69430914124502981247248948137, 5.89463443287140223083459360678, 6.01834080393432966584567233959, 6.02856331054450554983654699819, 6.16801099675516020279824416309, 6.86649025843642921970021235882, 6.97670638188966200650886100836, 7.14254561741020520523290292026, 7.43258593177356800160634817136, 7.66981501816775883330459380997, 7.84232125263057793666246361886

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