Properties

Degree 4
Conductor 1051
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 2·7-s − 3·8-s + 2·9-s + 10-s − 2·13-s + 2·14-s + 16-s − 2·17-s − 2·18-s + 6·19-s − 20-s − 4·23-s + 25-s + 2·26-s − 2·28-s − 29-s + 8·31-s + 32-s + 2·34-s + 2·35-s + 2·36-s − 8·37-s − 6·38-s + 3·40-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 1.06·8-s + 2/3·9-s + 0.316·10-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 1.37·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.185·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.338·35-s + 1/3·36-s − 1.31·37-s − 0.973·38-s + 0.474·40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1051 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1051 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1051\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1051} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1051,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3645581172$
$L(\frac12)$  $\approx$  $0.3645581172$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 1051$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 1051$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad1051$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 28 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
23$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 9 T + 98 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T - 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 11 T + 186 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−19.9192197157, −19.2816849154, −18.7634524353, −18.3881418626, −17.614958913, −17.3764249297, −16.3810656411, −15.9473242149, −15.5499694778, −15.1295810711, −14.0481218242, −13.6628321476, −12.6248614085, −12.2079845591, −11.6700797126, −10.8589509119, −9.96803258695, −9.61521048704, −8.87365930634, −8.02656767372, −7.16998104239, −6.6129675061, −5.55039481587, −4.16802503134, −2.83796791922, 2.83796791922, 4.16802503134, 5.55039481587, 6.6129675061, 7.16998104239, 8.02656767372, 8.87365930634, 9.61521048704, 9.96803258695, 10.8589509119, 11.6700797126, 12.2079845591, 12.6248614085, 13.6628321476, 14.0481218242, 15.1295810711, 15.5499694778, 15.9473242149, 16.3810656411, 17.3764249297, 17.614958913, 18.3881418626, 18.7634524353, 19.2816849154, 19.9192197157

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