Properties

Degree 4
Conductor 5501
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 6·7-s − 3·8-s + 3·9-s + 9·10-s − 12·12-s − 6·13-s + 18·14-s + 9·15-s + 3·16-s − 17-s − 9·18-s − 12·20-s + 18·21-s + 3·23-s + 9·24-s − 2·25-s + 18·26-s − 24·28-s − 3·29-s − 27·30-s − 6·31-s − 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 9-s + 2.84·10-s − 3.46·12-s − 1.66·13-s + 4.81·14-s + 2.32·15-s + 3/4·16-s − 0.242·17-s − 2.12·18-s − 2.68·20-s + 3.92·21-s + 0.625·23-s + 1.83·24-s − 2/5·25-s + 3.53·26-s − 4.53·28-s − 0.557·29-s − 4.92·30-s − 1.07·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5501 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5501 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5501\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5501} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 5501,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5501$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 5501$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5501$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
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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

−17.8022839541, −17.2109707906, −17.0678969544, −16.5213046446, −16.2108320826, −15.9166043737, −15.0356352162, −14.8539849988, −13.4301771076, −12.9442109961, −12.2376946517, −11.9697120649, −11.5279109374, −10.7354247474, −10.337135515, −9.77830243237, −9.29002886087, −8.87686756526, −7.79981719344, −7.37938449659, −6.85118815429, −6.05266556184, −5.42419577499, −4.16314320374, −3.05339871431, 0, 0, 3.05339871431, 4.16314320374, 5.42419577499, 6.05266556184, 6.85118815429, 7.37938449659, 7.79981719344, 8.87686756526, 9.29002886087, 9.77830243237, 10.337135515, 10.7354247474, 11.5279109374, 11.9697120649, 12.2376946517, 12.9442109961, 13.4301771076, 14.8539849988, 15.0356352162, 15.9166043737, 16.2108320826, 16.5213046446, 17.0678969544, 17.2109707906, 17.8022839541

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