Properties

Degree 4
Conductor $ 3 \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  − 2·2-s − 3·3-s + 6·6-s − 6·7-s + 4·8-s + 4·9-s + 4·11-s − 2·13-s + 12·14-s − 4·16-s − 4·17-s − 8·18-s + 18·21-s − 8·22-s − 4·23-s − 12·24-s − 25-s + 4·26-s + 4·29-s − 10·31-s − 12·33-s + 8·34-s + 2·37-s + 6·39-s − 6·41-s − 36·42-s − 2·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 2.44·6-s − 2.26·7-s + 1.41·8-s + 4/3·9-s + 1.20·11-s − 0.554·13-s + 3.20·14-s − 16-s − 0.970·17-s − 1.88·18-s + 3.92·21-s − 1.70·22-s − 0.834·23-s − 2.44·24-s − 1/5·25-s + 0.784·26-s + 0.742·29-s − 1.79·31-s − 2.08·33-s + 1.37·34-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 5.55·42-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1083\)    =    \(3 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1083} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1083,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
19$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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−19.6970630576, −19.5490432498, −18.9344925121, −18.1213987035, −18.0285451209, −17.4264315599, −16.7022471733, −16.5806200716, −16.2562118377, −15.3791583737, −14.5011418501, −13.3514918073, −13.2593691405, −12.1048029115, −12.0629600164, −11.0017900091, −10.2432044592, −9.91407057019, −9.18016854907, −8.83642738145, −7.47848778675, −6.41173993137, −6.39084306103, −5.03912355415, −3.81591641511, 0, 3.81591641511, 5.03912355415, 6.39084306103, 6.41173993137, 7.47848778675, 8.83642738145, 9.18016854907, 9.91407057019, 10.2432044592, 11.0017900091, 12.0629600164, 12.1048029115, 13.2593691405, 13.3514918073, 14.5011418501, 15.3791583737, 16.2562118377, 16.5806200716, 16.7022471733, 17.4264315599, 18.0285451209, 18.1213987035, 18.9344925121, 19.5490432498, 19.6970630576

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