Properties

Degree 4
Conductor $ 23 \cdot 281 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4-s − 6·5-s + 6·6-s − 2·7-s + 3·9-s + 12·10-s − 3·11-s − 3·12-s + 2·13-s + 4·14-s + 18·15-s + 16-s − 6·18-s − 7·19-s − 6·20-s + 6·21-s + 6·22-s − 4·23-s + 19·25-s − 4·26-s − 2·28-s − 7·29-s − 36·30-s + 2·31-s + 2·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.68·5-s + 2.44·6-s − 0.755·7-s + 9-s + 3.79·10-s − 0.904·11-s − 0.866·12-s + 0.554·13-s + 1.06·14-s + 4.64·15-s + 1/4·16-s − 1.41·18-s − 1.60·19-s − 1.34·20-s + 1.30·21-s + 1.27·22-s − 0.834·23-s + 19/5·25-s − 0.784·26-s − 0.377·28-s − 1.29·29-s − 6.57·30-s + 0.359·31-s + 0.353·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6463 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6463 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6463\)    =    \(23 \cdot 281\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6463} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 6463,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{23,\;281\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{23,\;281\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
281$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 10 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 25 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 53 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T - 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 59 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 2 T - 97 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 36 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−17.6186763168, −17.1931468375, −16.7151695933, −16.1870190696, −15.9653502553, −15.4770160615, −15.0984645548, −14.3319841629, −13.1626295233, −12.7443322007, −12.2402495124, −11.7463989779, −11.252034775, −10.9657080292, −10.4310339424, −9.75599058364, −8.87199751955, −8.38650133656, −7.86817900905, −7.43493560283, −6.44883541202, −6.0168037044, −4.89778956715, −4.16294537091, −3.33337496158, 0, 0, 3.33337496158, 4.16294537091, 4.89778956715, 6.0168037044, 6.44883541202, 7.43493560283, 7.86817900905, 8.38650133656, 8.87199751955, 9.75599058364, 10.4310339424, 10.9657080292, 11.252034775, 11.7463989779, 12.2402495124, 12.7443322007, 13.1626295233, 14.3319841629, 15.0984645548, 15.4770160615, 15.9653502553, 16.1870190696, 16.7151695933, 17.1931468375, 17.6186763168

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