Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{6} \cdot 5^{2} $
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 + 4-s + 3·5-s − 2·7-s − 8-s − 3·10-s − 6·11-s + 2·14-s + 16-s + 3·20-s + 6·22-s + 4·25-s − 2·28-s − 32-s − 6·35-s − 3·40-s − 20·43-s − 6·44-s − 11·49-s − 4·50-s + 18·53-s − 18·55-s + 2·56-s + 24·59-s + 16·61-s + 64-s + 28·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.534·14-s + 1/4·16-s + 0.670·20-s + 1.27·22-s + 4/5·25-s − 0.377·28-s − 0.176·32-s − 1.01·35-s − 0.474·40-s − 3.04·43-s − 0.904·44-s − 1.57·49-s − 0.565·50-s + 2.47·53-s − 2.42·55-s + 0.267·56-s + 3.12·59-s + 2.04·61-s + 1/8·64-s + 3.42·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(583200\)    =    \(2^{5} \cdot 3^{6} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{583200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 583200,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.143786255$
$L(\frac12)$  $\approx$  $1.143786255$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3 \( 1 \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.310249526893034803263699830740, −8.268048909391662018895990050852, −7.58907033734171805431789545424, −6.84593942818273157797656705828, −6.73860900674537002760989556093, −6.35572304303649648480081327455, −5.38354729541915242732826157737, −5.37363015537505503550836847886, −5.13499928761574618124626716865, −3.92861197755400275794059610293, −3.48774792613401599337251547075, −2.58391653549265790674050070138, −2.44850978333263328730546301919, −1.72580411383323342691331766149, −0.58944949786391246601175874957, 0.58944949786391246601175874957, 1.72580411383323342691331766149, 2.44850978333263328730546301919, 2.58391653549265790674050070138, 3.48774792613401599337251547075, 3.92861197755400275794059610293, 5.13499928761574618124626716865, 5.37363015537505503550836847886, 5.38354729541915242732826157737, 6.35572304303649648480081327455, 6.73860900674537002760989556093, 6.84593942818273157797656705828, 7.58907033734171805431789545424, 8.268048909391662018895990050852, 8.310249526893034803263699830740

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