Properties

Degree 8
Conductor $ 2^{4} \cdot 3^{4} \cdot 5^{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  − 2-s + 3-s + 5·5-s − 6-s + 6·7-s − 5·10-s − 5·11-s + 12·13-s − 6·14-s + 5·15-s + 6·17-s − 16·19-s + 6·21-s + 5·22-s − 10·23-s + 10·25-s − 12·26-s + 6·29-s − 5·30-s − 3·31-s + 32-s − 5·33-s − 6·34-s + 30·35-s − 8·37-s + 16·38-s + 12·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 2.23·5-s − 0.408·6-s + 2.26·7-s − 1.58·10-s − 1.50·11-s + 3.32·13-s − 1.60·14-s + 1.29·15-s + 1.45·17-s − 3.67·19-s + 1.30·21-s + 1.06·22-s − 2.08·23-s + 2·25-s − 2.35·26-s + 1.11·29-s − 0.912·30-s − 0.538·31-s + 0.176·32-s − 0.870·33-s − 1.02·34-s + 5.07·35-s − 1.31·37-s + 2.59·38-s + 1.92·39-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{4} \cdot 3^{4} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.87615$
$L(\frac12)$  $\approx$  $1.87615$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) 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$C_4$ \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
good7$D_{4}$ \( ( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2:C_4$ \( 1 + 5 T - T^{2} - 5 p T^{3} - 184 T^{4} - 5 p^{2} T^{5} - p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 12 T + 51 T^{2} - 76 T^{3} + 9 T^{4} - 76 p T^{5} + 51 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 - 6 T - T^{2} + 18 T^{3} + 169 T^{4} + 18 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 578 p T^{5} + 117 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 200 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_4\times C_2$ \( 1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 288 p T^{5} + 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_4$$\times$$C_4$ \( ( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
37$C_4\times C_2$ \( 1 + 8 T + 27 T^{2} - 80 T^{3} - 1639 T^{4} - 80 p T^{5} + 27 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 786 p T^{5} + 95 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 - 20 T + 113 T^{2} + 270 T^{3} - 5851 T^{4} + 270 p T^{5} + 113 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 16 T + p T^{2} - 200 T^{3} - 1759 T^{4} - 200 p T^{5} + p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 5 T - 49 T^{2} + 5 p T^{3} + 1736 T^{4} + 5 p^{2} T^{5} - 49 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 410 p T^{5} - 21 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 - 16 T + 29 T^{2} + 868 T^{3} - 9191 T^{4} + 868 p T^{5} + 29 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 20 T + 89 T^{2} - 20 p T^{3} - 22639 T^{4} - 20 p^{2} T^{5} + 89 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 2 T + 51 T^{2} - 376 T^{3} + 6389 T^{4} - 376 p T^{5} + 51 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 2 T - 55 T^{2} + 578 T^{3} + 3679 T^{4} + 578 p T^{5} - 55 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 13 T - 19 T^{2} - 951 T^{3} - 6196 T^{4} - 951 p T^{5} - 19 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 - 2 T + 35 T^{2} - 632 T^{3} + 8789 T^{4} - 632 p T^{5} + 35 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 7 T - 73 T^{2} + 835 T^{3} + 1816 T^{4} + 835 p T^{5} - 73 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
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

−9.734748605561883321750643071433, −8.937730417884018014575819450147, −8.920591426148380215598855230777, −8.829523272565290399427249198699, −8.466324681271482704732631883474, −8.235068105895055738684134390160, −7.987934348601374212507366620492, −7.977295075985077516107503227531, −7.76082225247954562434367029336, −6.79730757470107264759183094159, −6.65649491396160214640373799445, −6.44167096348144035519339700530, −6.03746520332227722150304872891, −5.82372521720221727346789161515, −5.44651566070203651198041074908, −5.40866165873349448291739280325, −4.78863267738919922204329203112, −4.46643273137775871687519530667, −4.09191407233855373180184861887, −3.64375192631268278900769211989, −3.17409085559959444532726007576, −2.54035898568887869145233121268, −1.84506009937836413901459379786, −1.77705325616513407938804107169, −1.60803581944367060453600804459, 1.60803581944367060453600804459, 1.77705325616513407938804107169, 1.84506009937836413901459379786, 2.54035898568887869145233121268, 3.17409085559959444532726007576, 3.64375192631268278900769211989, 4.09191407233855373180184861887, 4.46643273137775871687519530667, 4.78863267738919922204329203112, 5.40866165873349448291739280325, 5.44651566070203651198041074908, 5.82372521720221727346789161515, 6.03746520332227722150304872891, 6.44167096348144035519339700530, 6.65649491396160214640373799445, 6.79730757470107264759183094159, 7.76082225247954562434367029336, 7.977295075985077516107503227531, 7.987934348601374212507366620492, 8.235068105895055738684134390160, 8.466324681271482704732631883474, 8.829523272565290399427249198699, 8.920591426148380215598855230777, 8.937730417884018014575819450147, 9.734748605561883321750643071433

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