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-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-s + ⋯
L(s)  = 1  + 2-s − 2·3-s − 2·6-s − 8-s + 3·9-s − 2·11-s − 16-s − 4·17-s + 3·18-s − 2·19-s − 2·22-s + 2·24-s − 4·27-s + 4·33-s − 4·34-s − 2·38-s + 41-s − 43-s + 2·48-s − 49-s + 8·51-s − 4·54-s + 4·57-s + 59-s + 64-s + 4·66-s − 67-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  =  $(4,\ 3240000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.07231841195$
$L(\frac12)$  $\approx$  $0.07231841195$
$L(1)$   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 4. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - T + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$ \( ( 1 + T )^{4} \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
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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

−9.910023944342792120838150622733, −9.221672433493689494542748373485, −8.931788012563920906567232087794, −8.484988356238490620612424190935, −8.092650685060006443180412257156, −7.52731592307891002553681013779, −6.91336963670413770661341706546, −6.62827361643327361832839757443, −6.42028049841365022151502090050, −6.10830087856580903077164341736, −5.36970526932147481326199900939, −5.31283448836759103443258226680, −4.60099810867796520080950289180, −4.58211161765208124931401182762, −4.18826557465494761172551114441, −3.72573923948335484107591889285, −2.53060944909435527588667559895, −2.44656023537320876925877225840, −1.77606184994408508858255127027, −0.18068456797024635616045494218, 0.18068456797024635616045494218, 1.77606184994408508858255127027, 2.44656023537320876925877225840, 2.53060944909435527588667559895, 3.72573923948335484107591889285, 4.18826557465494761172551114441, 4.58211161765208124931401182762, 4.60099810867796520080950289180, 5.31283448836759103443258226680, 5.36970526932147481326199900939, 6.10830087856580903077164341736, 6.42028049841365022151502090050, 6.62827361643327361832839757443, 6.91336963670413770661341706546, 7.52731592307891002553681013779, 8.092650685060006443180412257156, 8.484988356238490620612424190935, 8.931788012563920906567232087794, 9.221672433493689494542748373485, 9.910023944342792120838150622733

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