Properties

Degree 4
Conductor 6229
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 − 2·3-s + 4·4-s − 6·5-s + 6·6-s − 2·7-s − 3·8-s − 2·9-s + 18·10-s + 11-s − 8·12-s − 4·13-s + 6·14-s + 12·15-s + 3·16-s − 6·17-s + 6·18-s + 19-s − 24·20-s + 4·21-s − 3·22-s − 6·23-s + 6·24-s + 18·25-s + 12·26-s + 10·27-s − 8·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 2.68·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s + 5.69·10-s + 0.301·11-s − 2.30·12-s − 1.10·13-s + 1.60·14-s + 3.09·15-s + 3/4·16-s − 1.45·17-s + 1.41·18-s + 0.229·19-s − 5.36·20-s + 0.872·21-s − 0.639·22-s − 1.25·23-s + 1.22·24-s + 18/5·25-s + 2.35·26-s + 1.92·27-s − 1.51·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6229 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6229 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6229\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6229} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 6229,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 6229$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 6229$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad6229$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 125 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T - 5 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 82 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T - 4 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 12 T + 99 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 73 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T - 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T - 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 109 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−17.514180109, −17.3704096896, −16.835320934, −16.3896426878, −15.8503021612, −15.6922102646, −15.0434213425, −14.4433970564, −13.6107791985, −12.4776076382, −12.0769496245, −11.9278389573, −11.2528360359, −10.9608073948, −10.2692818627, −9.6719976273, −8.89462827453, −8.46046702832, −8.05478098701, −7.39652759875, −6.84614693268, −6.10804533723, −4.93601810327, −4.09462780235, −3.06355765592, 0, 0, 3.06355765592, 4.09462780235, 4.93601810327, 6.10804533723, 6.84614693268, 7.39652759875, 8.05478098701, 8.46046702832, 8.89462827453, 9.6719976273, 10.2692818627, 10.9608073948, 11.2528360359, 11.9278389573, 12.0769496245, 12.4776076382, 13.6107791985, 14.4433970564, 15.0434213425, 15.6922102646, 15.8503021612, 16.3896426878, 16.835320934, 17.3704096896, 17.514180109

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