Properties

Degree 4
Conductor $ 2 \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s − 5-s − 2·7-s + 2·8-s − 9-s − 3·11-s + 12-s + 4·13-s + 15-s + 16-s − 4·17-s + 3·19-s + 20-s + 2·21-s + 5·23-s − 2·24-s + 25-s + 2·28-s + 3·29-s − 4·31-s − 4·32-s + 3·33-s + 2·35-s + 36-s + 2·37-s − 4·39-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.707·8-s − 1/3·9-s − 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.688·19-s + 0.223·20-s + 0.436·21-s + 1.04·23-s − 0.408·24-s + 1/5·25-s + 0.377·28-s + 0.557·29-s − 0.718·31-s − 0.707·32-s + 0.522·33-s + 0.338·35-s + 1/6·36-s + 0.328·37-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(862\)    =    \(2 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{862} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 862,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3738532046$
$L(\frac12)$  $\approx$  $0.3738532046$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;431\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;431\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
431$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 16 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$D_{4}$ \( 1 - 5 T + 38 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T - 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 68 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
61$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−19.7765939928, −19.0919369616, −18.4061727367, −18.2127234337, −17.3956977529, −16.8155298665, −16.2785456546, −15.7323045279, −15.3149141682, −14.3171558059, −13.6578953069, −13.0761180034, −12.7716707719, −11.7429421184, −11.0014646541, −10.7410312051, −9.75009760266, −8.97875057169, −8.22716555782, −7.36438531164, −6.45183254625, −5.48872358295, −4.58085723267, −3.30963911267, 3.30963911267, 4.58085723267, 5.48872358295, 6.45183254625, 7.36438531164, 8.22716555782, 8.97875057169, 9.75009760266, 10.7410312051, 11.0014646541, 11.7429421184, 12.7716707719, 13.0761180034, 13.6578953069, 14.3171558059, 15.3149141682, 15.7323045279, 16.2785456546, 16.8155298665, 17.3956977529, 18.2127234337, 18.4061727367, 19.0919369616, 19.7765939928

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