Properties

Degree 4
Conductor $ 2 \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·7-s + 8-s − 2·9-s + 3·11-s − 2·14-s + 3·16-s + 2·18-s − 3·22-s + 6·23-s + 2·25-s − 2·28-s + 3·29-s − 3·32-s + 2·36-s + 13·37-s − 11·43-s − 3·44-s − 6·46-s − 3·49-s − 2·50-s − 3·53-s + 2·56-s − 3·58-s − 4·63-s − 5·64-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.534·14-s + 3/4·16-s + 0.471·18-s − 0.639·22-s + 1.25·23-s + 2/5·25-s − 0.377·28-s + 0.557·29-s − 0.530·32-s + 1/3·36-s + 2.13·37-s − 1.67·43-s − 0.452·44-s − 0.884·46-s − 3/7·49-s − 0.282·50-s − 0.412·53-s + 0.267·56-s − 0.393·58-s − 0.503·63-s − 5/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11858\)    =    \(2 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11858} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11858,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6681225335$
$L(\frac12)$  $\approx$  $0.6681225335$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
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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

−11.16679809343792126919038274667, −10.96555341410425636086309997597, −10.08017141377105238169731944446, −9.581168449578710297688473560356, −9.096303062908669530462883454538, −8.574223034598092966737460294096, −8.149624255490186637541192452864, −7.57660400870567455681765648785, −6.67999960541019473841907537703, −6.13817826308417906229670655342, −5.18812111025287882801104414890, −4.71839188611404283173472155542, −3.75968339734596524308915012701, −2.79882649186254505734959643146, −1.23396492719572603792206010777, 1.23396492719572603792206010777, 2.79882649186254505734959643146, 3.75968339734596524308915012701, 4.71839188611404283173472155542, 5.18812111025287882801104414890, 6.13817826308417906229670655342, 6.67999960541019473841907537703, 7.57660400870567455681765648785, 8.149624255490186637541192452864, 8.574223034598092966737460294096, 9.096303062908669530462883454538, 9.581168449578710297688473560356, 10.08017141377105238169731944446, 10.96555341410425636086309997597, 11.16679809343792126919038274667

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