Properties

Degree 4
Conductor $ 2^{2} \cdot 97 $
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-s − 2·3-s − 2·5-s + 2·6-s + 7-s + 8-s + 9-s + 2·10-s − 2·11-s + 4·13-s − 14-s + 4·15-s − 16-s + 3·17-s − 18-s − 4·19-s − 2·21-s + 2·22-s − 23-s − 2·24-s − 4·26-s − 2·27-s − 4·30-s − 31-s + 4·33-s − 3·34-s − 2·35-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1.03·15-s − 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.436·21-s + 0.426·22-s − 0.208·23-s − 0.408·24-s − 0.784·26-s − 0.384·27-s − 0.730·30-s − 0.179·31-s + 0.696·33-s − 0.514·34-s − 0.338·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(388\)    =    \(2^{2} \cdot 97\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{388} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 388,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1982006890$
$L(\frac12)$  $\approx$  $0.1982006890$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;97\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;97\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 14 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 6 T - 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 10 T + 124 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−19.9804003514, −19.0844364656, −18.7088738959, −18.2748943046, −17.3626593031, −17.2067930848, −16.394289453, −15.8395495108, −15.3299673196, −14.4018758175, −13.5620147753, −12.7012057581, −12.0306318929, −11.2513554638, −10.890046816, −10.1528184391, −9.00839466888, −8.16229040744, −7.55108787382, −6.28409327055, −5.34389401817, −4.02372308969, 4.02372308969, 5.34389401817, 6.28409327055, 7.55108787382, 8.16229040744, 9.00839466888, 10.1528184391, 10.890046816, 11.2513554638, 12.0306318929, 12.7012057581, 13.5620147753, 14.4018758175, 15.3299673196, 15.8395495108, 16.394289453, 17.2067930848, 17.3626593031, 18.2748943046, 18.7088738959, 19.0844364656, 19.9804003514

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