Properties

Degree 4
Conductor 461
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s + 5-s + 3·9-s + 2·11-s + 6·12-s + 13-s − 3·15-s + 17-s − 19-s − 2·20-s − 3·23-s + 2·29-s − 4·31-s − 6·33-s − 6·36-s − 3·39-s + 16·41-s − 6·43-s − 4·44-s + 3·45-s − 4·47-s − 6·49-s − 3·51-s − 2·52-s + 53-s + 2·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 0.447·5-s + 9-s + 0.603·11-s + 1.73·12-s + 0.277·13-s − 0.774·15-s + 0.242·17-s − 0.229·19-s − 0.447·20-s − 0.625·23-s + 0.371·29-s − 0.718·31-s − 1.04·33-s − 36-s − 0.480·39-s + 2.49·41-s − 0.914·43-s − 0.603·44-s + 0.447·45-s − 0.583·47-s − 6/7·49-s − 0.420·51-s − 0.277·52-s + 0.137·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 461 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 461 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(461\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{461} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 461,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2458864264$
$L(\frac12)$  $\approx$  $0.2458864264$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 461$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 461$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad461$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 13 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 17 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 5 T - 7 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 5 T + 106 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 10 T + 124 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 44 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 5 T + 145 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.5818249319, −18.8664797957, −18.1284338381, −17.8116839057, −17.4484695511, −16.7423739165, −16.3619168852, −15.5940585516, −14.4257598969, −14.1852060086, −13.1711538133, −12.7237105692, −11.8277861803, −11.3431082453, −10.6307297187, −9.74673643577, −9.11211751757, −8.10584670363, −6.73127813828, −5.94451781406, −5.24079847106, −4.16569729781, 4.16569729781, 5.24079847106, 5.94451781406, 6.73127813828, 8.10584670363, 9.11211751757, 9.74673643577, 10.6307297187, 11.3431082453, 11.8277861803, 12.7237105692, 13.1711538133, 14.1852060086, 14.4257598969, 15.5940585516, 16.3619168852, 16.7423739165, 17.4484695511, 17.8116839057, 18.1284338381, 18.8664797957, 19.5818249319

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