Properties

Degree 4
Conductor $ 2^{2} \cdot 7 \cdot 23 $
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 − 3·7-s + 2·9-s + 2·11-s + 2·12-s + 2·13-s + 16-s − 2·17-s + 6·21-s + 3·23-s − 4·25-s − 6·27-s + 3·28-s + 8·29-s − 2·31-s − 4·33-s − 2·36-s − 6·37-s − 4·39-s + 2·41-s + 2·43-s − 2·44-s + 4·47-s − 2·48-s + 4·51-s − 2·52-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 1.13·7-s + 2/3·9-s + 0.603·11-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.30·21-s + 0.625·23-s − 4/5·25-s − 1.15·27-s + 0.566·28-s + 1.48·29-s − 0.359·31-s − 0.696·33-s − 1/3·36-s − 0.986·37-s − 0.640·39-s + 0.312·41-s + 0.304·43-s − 0.301·44-s + 0.583·47-s − 0.288·48-s + 0.560·51-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{644} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 644,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3087021439$
$L(\frac12)$  $\approx$  $0.3087021439$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
31$D_{4}$ \( 1 + 2 T - 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
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

−19.423796309, −19.1492733656, −18.4313169827, −17.7791037642, −17.3057642744, −16.8749677391, −16.1071861552, −15.7309621332, −15.0375848666, −13.9804591873, −13.5544476283, −12.787605563, −12.265325203, −11.560327967, −10.854458304, −10.1370351403, −9.37368605065, −8.74377833705, −7.48523119168, −6.45298028941, −6.00213247982, −4.82710342129, −3.62441303496, 3.62441303496, 4.82710342129, 6.00213247982, 6.45298028941, 7.48523119168, 8.74377833705, 9.37368605065, 10.1370351403, 10.854458304, 11.560327967, 12.265325203, 12.787605563, 13.5544476283, 13.9804591873, 15.0375848666, 15.7309621332, 16.1071861552, 16.8749677391, 17.3057642744, 17.7791037642, 18.4313169827, 19.1492733656, 19.423796309

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