Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 2·13-s + 2·19-s + 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s − 4·91-s + 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·7-s + 2·13-s + 2·19-s + 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s − 4·91-s + 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3240000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.123231962$
$L(\frac12)$  $\approx$  $1.123231962$
$L(1)$   not available
$L(1)$   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−9.511175960404781618649421398952, −9.341590092977878629828543916595, −9.054463240882008076704081073943, −8.488175431717945062313005350818, −8.102257043226013423307724383503, −7.77444311268645063685427607118, −7.17833273073807590623599083198, −6.79555112695010867992420096465, −6.47279737514731729846813250396, −6.12224649955494156101339490121, −5.67741807178035927669926686561, −5.47907828137923836687614048949, −4.67081895796964342886463253640, −4.19746307343099414941982481620, −3.54627576406584498228687204239, −3.50745555893537077493993703579, −2.79957114848715855936262809137, −2.60133102228583573717590057644, −1.35574886675700468075506851066, −0.927879736999085279198485373471, 0.927879736999085279198485373471, 1.35574886675700468075506851066, 2.60133102228583573717590057644, 2.79957114848715855936262809137, 3.50745555893537077493993703579, 3.54627576406584498228687204239, 4.19746307343099414941982481620, 4.67081895796964342886463253640, 5.47907828137923836687614048949, 5.67741807178035927669926686561, 6.12224649955494156101339490121, 6.47279737514731729846813250396, 6.79555112695010867992420096465, 7.17833273073807590623599083198, 7.77444311268645063685427607118, 8.102257043226013423307724383503, 8.488175431717945062313005350818, 9.054463240882008076704081073943, 9.341590092977878629828543916595, 9.511175960404781618649421398952

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