Properties

Degree 4
Conductor 62563
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 7·7-s − 3·8-s + 4·9-s + 12·10-s − 4·11-s − 12·12-s − 4·13-s + 21·14-s + 12·15-s + 3·16-s − 9·17-s − 12·18-s + 2·19-s − 16·20-s + 21·21-s + 12·22-s + 9·24-s + 6·25-s + 12·26-s − 6·27-s − 28·28-s − 2·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 2.64·7-s − 1.06·8-s + 4/3·9-s + 3.79·10-s − 1.20·11-s − 3.46·12-s − 1.10·13-s + 5.61·14-s + 3.09·15-s + 3/4·16-s − 2.18·17-s − 2.82·18-s + 0.458·19-s − 3.57·20-s + 4.58·21-s + 2.55·22-s + 1.83·24-s + 6/5·25-s + 2.35·26-s − 1.15·27-s − 5.29·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62563 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62563 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(62563\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{62563} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 62563,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 62563$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 62563$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad62563$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 120 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 17 T + 131 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 94 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 32 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 244 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 13 T + 223 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−15.4885081690, −15.2265244062, −14.7119383576, −13.3606050657, −13.1065499074, −12.8591722306, −12.3100966576, −11.7294381243, −11.4971211104, −11.0203820692, −10.4583757227, −10.1853925163, −9.61297224984, −9.37373654569, −8.70589242499, −8.29251791678, −7.52616316265, −7.05729989084, −6.95123042819, −6.27356720529, −5.43938462819, −5.08635742575, −3.84853042719, −3.52909306076, −2.37821411186, 0, 0, 0, 2.37821411186, 3.52909306076, 3.84853042719, 5.08635742575, 5.43938462819, 6.27356720529, 6.95123042819, 7.05729989084, 7.52616316265, 8.29251791678, 8.70589242499, 9.37373654569, 9.61297224984, 10.1853925163, 10.4583757227, 11.0203820692, 11.4971211104, 11.7294381243, 12.3100966576, 12.8591722306, 13.1065499074, 13.3606050657, 14.7119383576, 15.2265244062, 15.4885081690

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