Properties

Degree 4
Conductor 5641
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 − 4·3-s + 2·4-s − 5·5-s + 8·6-s − 4·7-s − 4·8-s + 8·9-s + 10·10-s − 11-s − 8·12-s + 8·14-s + 20·15-s + 8·16-s + 4·17-s − 16·18-s − 3·19-s − 10·20-s + 16·21-s + 2·22-s − 9·23-s + 16·24-s + 10·25-s − 12·27-s − 8·28-s − 10·29-s − 40·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s − 2.23·5-s + 3.26·6-s − 1.51·7-s − 1.41·8-s + 8/3·9-s + 3.16·10-s − 0.301·11-s − 2.30·12-s + 2.13·14-s + 5.16·15-s + 2·16-s + 0.970·17-s − 3.77·18-s − 0.688·19-s − 2.23·20-s + 3.49·21-s + 0.426·22-s − 1.87·23-s + 3.26·24-s + 2·25-s − 2.30·27-s − 1.51·28-s − 1.85·29-s − 7.30·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5641\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5641} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 5641,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5641$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 5641$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5641$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 70 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 244 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 3 T - 101 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−17.8716348883, −17.2408286174, −16.7155510117, −16.5208175795, −16.1581989797, −15.6468185365, −15.2114122239, −14.7918784345, −13.4108734249, −12.6366758466, −12.2716210805, −11.9292597246, −11.5334049858, −11.079017939, −10.4234379986, −9.83438859301, −9.46707132659, −8.34400562979, −7.95561643441, −7.27771121984, −6.55985046799, −5.97259022683, −5.44115532946, −4.04092722488, −3.45992068722, 0, 0, 3.45992068722, 4.04092722488, 5.44115532946, 5.97259022683, 6.55985046799, 7.27771121984, 7.95561643441, 8.34400562979, 9.46707132659, 9.83438859301, 10.4234379986, 11.079017939, 11.5334049858, 11.9292597246, 12.2716210805, 12.6366758466, 13.4108734249, 14.7918784345, 15.2114122239, 15.6468185365, 16.1581989797, 16.5208175795, 16.7155510117, 17.2408286174, 17.8716348883

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