Properties

Degree 4
Conductor $ 2^{5} \cdot 37 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·7-s − 9-s + 11-s − 2·17-s + 3·21-s + 6·23-s − 6·25-s + 4·29-s + 4·31-s − 33-s + 5·37-s + 41-s − 2·43-s − 3·47-s + 3·49-s + 2·51-s + 3·53-s − 4·59-s + 6·61-s + 3·63-s − 4·67-s − 6·69-s + 11·71-s − 23·73-s + 6·75-s − 3·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s − 1/3·9-s + 0.301·11-s − 0.485·17-s + 0.654·21-s + 1.25·23-s − 6/5·25-s + 0.742·29-s + 0.718·31-s − 0.174·33-s + 0.821·37-s + 0.156·41-s − 0.304·43-s − 0.437·47-s + 3/7·49-s + 0.280·51-s + 0.412·53-s − 0.520·59-s + 0.768·61-s + 0.377·63-s − 0.488·67-s − 0.722·69-s + 1.30·71-s − 2.69·73-s + 0.692·75-s − 0.341·77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1184\)    =    \(2^{5} \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1184} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1184,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4447513757$
$L(\frac12)$  $\approx$  $0.4447513757$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;37\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;37\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 23 T + 264 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 19 T + 194 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−19.63395993, −19.173860191, −18.7811477897, −17.8432313014, −17.5089150259, −16.9774712084, −16.338108163, −15.9434029607, −15.2519616686, −14.7226698649, −13.8445619238, −13.2965100208, −12.8288252272, −11.9713749254, −11.5695433336, −10.8318004249, −10.0762974332, −9.45681634241, −8.77884882311, −7.83538718435, −6.80484545334, −6.28666869558, −5.42977672285, −4.24669402094, −2.95555289946, 2.95555289946, 4.24669402094, 5.42977672285, 6.28666869558, 6.80484545334, 7.83538718435, 8.77884882311, 9.45681634241, 10.0762974332, 10.8318004249, 11.5695433336, 11.9713749254, 12.8288252272, 13.2965100208, 13.8445619238, 14.7226698649, 15.2519616686, 15.9434029607, 16.338108163, 16.9774712084, 17.5089150259, 17.8432313014, 18.7811477897, 19.173860191, 19.63395993

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