Properties

Degree 4
Conductor $ 2^{2} \cdot 3 \cdot 59 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 5-s − 3·7-s + 2·11-s + 2·12-s − 2·15-s + 16-s + 4·17-s − 5·19-s − 20-s + 6·21-s + 8·23-s + 25-s + 5·27-s + 3·28-s − 5·29-s − 6·31-s − 4·33-s − 3·35-s + 2·37-s − 3·41-s + 2·43-s − 2·44-s − 2·47-s − 2·48-s + 3·49-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.577·12-s − 0.516·15-s + 1/4·16-s + 0.970·17-s − 1.14·19-s − 0.223·20-s + 1.30·21-s + 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.566·28-s − 0.928·29-s − 1.07·31-s − 0.696·33-s − 0.507·35-s + 0.328·37-s − 0.468·41-s + 0.304·43-s − 0.301·44-s − 0.291·47-s − 0.288·48-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(708\)    =    \(2^{2} \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{708} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 708,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3253436263$
$L(\frac12)$  $\approx$  $0.3253436263$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;59\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;59\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$V_4$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−19.2635183129, −18.8983865543, −18.3672695104, −17.5272060415, −17.1046203043, −16.7104638256, −16.3624662849, −15.3501628981, −14.6579725608, −14.1645025024, −13.0779421625, −12.8868227016, −12.2001214421, −11.3216041667, −10.7981771025, −9.97320924666, −9.2915682178, −8.67647726032, −7.32177052208, −6.39986997081, −5.82876143655, −4.9097746227, −3.42333274451, 3.42333274451, 4.9097746227, 5.82876143655, 6.39986997081, 7.32177052208, 8.67647726032, 9.2915682178, 9.97320924666, 10.7981771025, 11.3216041667, 12.2001214421, 12.8868227016, 13.0779421625, 14.1645025024, 14.6579725608, 15.3501628981, 16.3624662849, 16.7104638256, 17.1046203043, 17.5272060415, 18.3672695104, 18.8983865543, 19.2635183129

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