Properties

Degree 4
Conductor 54983
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s − 5·5-s + 6·6-s − 5·7-s + 4·8-s + 2·9-s + 10·10-s − 10·11-s + 10·14-s + 15·15-s − 4·16-s − 2·17-s − 4·18-s − 4·19-s + 15·21-s + 20·22-s + 2·23-s − 12·24-s + 11·25-s + 6·27-s − 8·29-s − 30·30-s − 8·31-s + 30·33-s + 4·34-s + 25·35-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 1.41·8-s + 2/3·9-s + 3.16·10-s − 3.01·11-s + 2.67·14-s + 3.87·15-s − 16-s − 0.485·17-s − 0.942·18-s − 0.917·19-s + 3.27·21-s + 4.26·22-s + 0.417·23-s − 2.44·24-s + 11/5·25-s + 1.15·27-s − 1.48·29-s − 5.47·30-s − 1.43·31-s + 5.22·33-s + 0.685·34-s + 4.22·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 54983 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 54983 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(54983\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{54983} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 54983,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 54983$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 54983$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad54983$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 176 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 11 T + 85 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$D_{4}$ \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T - 47 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 120 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T - 51 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−15.6517256201, −15.1514975049, −14.8236425639, −13.716633134, −13.0597375798, −12.967278024, −12.6368290865, −12.0867604761, −11.4167486141, −11.0577651374, −10.7283665737, −10.4289701573, −9.83992658689, −9.28767273356, −8.6791136623, −8.22951610227, −7.78980064228, −7.30578498626, −6.91342997451, −5.8964284844, −5.62495219491, −4.8514861056, −4.32310572595, −3.46315708887, −2.77777259775, 0, 0, 0, 2.77777259775, 3.46315708887, 4.32310572595, 4.8514861056, 5.62495219491, 5.8964284844, 6.91342997451, 7.30578498626, 7.78980064228, 8.22951610227, 8.6791136623, 9.28767273356, 9.83992658689, 10.4289701573, 10.7283665737, 11.0577651374, 11.4167486141, 12.0867604761, 12.6368290865, 12.967278024, 13.0597375798, 13.716633134, 14.8236425639, 15.1514975049, 15.6517256201

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