Properties

Degree 4
Conductor $ 3^{2} \cdot 11^{2} \cdot 19^{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  − 4·2-s − 3-s + 8·4-s + 4·6-s − 4·7-s − 8·8-s − 2·9-s − 8·12-s + 16·14-s − 4·16-s + 8·18-s + 4·21-s + 8·24-s − 9·25-s + 5·27-s − 32·28-s + 32·32-s − 16·36-s − 16·41-s − 16·42-s − 12·43-s + 4·48-s − 2·49-s + 36·50-s − 12·53-s − 20·54-s + 32·56-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 1.51·7-s − 2.82·8-s − 2/3·9-s − 2.30·12-s + 4.27·14-s − 16-s + 1.88·18-s + 0.872·21-s + 1.63·24-s − 9/5·25-s + 0.962·27-s − 6.04·28-s + 5.65·32-s − 8/3·36-s − 2.49·41-s − 2.46·42-s − 1.82·43-s + 0.577·48-s − 2/7·49-s + 5.09·50-s − 1.64·53-s − 2.72·54-s + 4.27·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(393129\)    =    \(3^{2} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{393129} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 393129,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.603539619290756001226038948684, −8.214596534850172998316450386324, −7.73556351389179885223258451389, −7.17115755169439685292874478446, −6.77654603285244896156727941052, −6.36261389471308870138602900888, −6.05894787659039661644210218582, −5.09690794065788751110696140894, −4.80674258105858635103949879256, −3.62924072940666698491701863630, −3.33280490374154022682414025774, −2.24326475114330249154243176400, −1.78787687456595466327825076524, −0.62260408018451229277893602459, 0, 0.62260408018451229277893602459, 1.78787687456595466327825076524, 2.24326475114330249154243176400, 3.33280490374154022682414025774, 3.62924072940666698491701863630, 4.80674258105858635103949879256, 5.09690794065788751110696140894, 6.05894787659039661644210218582, 6.36261389471308870138602900888, 6.77654603285244896156727941052, 7.17115755169439685292874478446, 7.73556351389179885223258451389, 8.214596534850172998316450386324, 8.603539619290756001226038948684

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