Properties

Degree 4
Conductor $ 2^{4} \cdot 7^{2} \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s − 2·9-s + 10·11-s + 16·23-s − 9·25-s − 4·29-s + 20·37-s + 2·43-s + 2·49-s − 8·53-s + 6·63-s − 24·67-s + 4·71-s − 30·77-s + 16·79-s − 5·81-s − 20·99-s + 4·107-s − 20·113-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1.13·7-s − 2/3·9-s + 3.01·11-s + 3.33·23-s − 9/5·25-s − 0.742·29-s + 3.28·37-s + 0.304·43-s + 2/7·49-s − 1.09·53-s + 0.755·63-s − 2.93·67-s + 0.474·71-s − 3.41·77-s + 1.80·79-s − 5/9·81-s − 2.01·99-s + 0.386·107-s − 1.88·113-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(283024\)    =    \(2^{4} \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{283024} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 283024,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.825864643$
$L(\frac12)$  $\approx$  $1.825864643$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−9.084561409834522310688102275122, −8.709339720534257240327886562953, −7.64552271968190118551531459773, −7.62429229152375892340957919668, −6.76474142604661213574836140046, −6.44323514092267885981017350010, −6.21156913138879588632301451179, −5.67409965982210077643645681036, −4.91419379168876950061294361671, −4.15537276955970084853193543262, −3.93593184874987401028815060034, −3.14091634253431942337825536555, −2.82511261070703297705797963037, −1.62016711898696823707962362202, −0.869557064463155631256543449078, 0.869557064463155631256543449078, 1.62016711898696823707962362202, 2.82511261070703297705797963037, 3.14091634253431942337825536555, 3.93593184874987401028815060034, 4.15537276955970084853193543262, 4.91419379168876950061294361671, 5.67409965982210077643645681036, 6.21156913138879588632301451179, 6.44323514092267885981017350010, 6.76474142604661213574836140046, 7.62429229152375892340957919668, 7.64552271968190118551531459773, 8.709339720534257240327886562953, 9.084561409834522310688102275122

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