Properties

Degree 4
Conductor $ 3^{2} \cdot 47^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·5-s − 6·7-s + 4·8-s + 9-s + 8·10-s − 8·11-s − 2·13-s + 12·14-s − 4·16-s + 2·17-s − 2·18-s − 12·19-s + 16·22-s + 12·23-s + 3·25-s + 4·26-s + 2·31-s − 4·34-s + 24·35-s + 2·37-s + 24·38-s − 16·40-s − 4·41-s − 6·43-s − 4·45-s − 24·46-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.78·5-s − 2.26·7-s + 1.41·8-s + 1/3·9-s + 2.52·10-s − 2.41·11-s − 0.554·13-s + 3.20·14-s − 16-s + 0.485·17-s − 0.471·18-s − 2.75·19-s + 3.41·22-s + 2.50·23-s + 3/5·25-s + 0.784·26-s + 0.359·31-s − 0.685·34-s + 4.05·35-s + 0.328·37-s + 3.89·38-s − 2.52·40-s − 0.624·41-s − 0.914·43-s − 0.596·45-s − 3.53·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(19881\)    =    \(3^{2} \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19881} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 19881,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;47\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
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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

−16.3084950271, −15.7159115639, −15.5159935893, −15.1680730202, −14.6489059291, −13.3552681657, −13.2982222373, −12.78597999, −12.6657932749, −11.9075398839, −11.064550122, −10.5981068261, −10.1950027975, −9.89655043622, −9.22229106146, −8.53725475056, −8.40526764581, −7.63249491007, −7.30736849694, −6.71008245044, −5.83486052932, −4.7534983042, −4.36466935125, −3.31847810115, −2.77908676833, 0, 0, 2.77908676833, 3.31847810115, 4.36466935125, 4.7534983042, 5.83486052932, 6.71008245044, 7.30736849694, 7.63249491007, 8.40526764581, 8.53725475056, 9.22229106146, 9.89655043622, 10.1950027975, 10.5981068261, 11.064550122, 11.9075398839, 12.6657932749, 12.78597999, 13.2982222373, 13.3552681657, 14.6489059291, 15.1680730202, 15.5159935893, 15.7159115639, 16.3084950271

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