Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{6} \cdot 5^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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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  =  1
Selberg data  =  $(4,\ 583200,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.175508045180459617652224529359, −7.82723843094092867969941089213, −7.07832589673054274134741987475, −6.71508725988629390136097161445, −6.50809047734617527891647867661, −6.10384069202259277792945056161, −5.26189135061292063008867608145, −4.77709333112371596512170551343, −4.38306493815359613482087430509, −3.72351464227480953738971596775, −3.38549178369413487864105189864, −3.17052507023677297081073811536, −2.00274833407885544645947603057, −1.28826140651909171078200157979, 0, 1.28826140651909171078200157979, 2.00274833407885544645947603057, 3.17052507023677297081073811536, 3.38549178369413487864105189864, 3.72351464227480953738971596775, 4.38306493815359613482087430509, 4.77709333112371596512170551343, 5.26189135061292063008867608145, 6.10384069202259277792945056161, 6.50809047734617527891647867661, 6.71508725988629390136097161445, 7.07832589673054274134741987475, 7.82723843094092867969941089213, 8.175508045180459617652224529359

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