Properties

Degree 4
Conductor $ 31 \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  − 2-s − 3-s + 4-s + 6-s + 3·7-s − 3·8-s − 9-s − 3·11-s − 12-s − 3·14-s + 16-s − 2·17-s + 18-s + 2·19-s − 3·21-s + 3·22-s + 3·24-s − 6·25-s + 3·28-s + 2·29-s − 31-s + 32-s + 3·33-s + 2·34-s − 36-s + 5·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.654·21-s + 0.639·22-s + 0.612·24-s − 6/5·25-s + 0.566·28-s + 0.371·29-s − 0.179·31-s + 0.176·32-s + 0.522·33-s + 0.342·34-s − 1/6·36-s + 0.821·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1147 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1147 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1147\)    =    \(31 \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1147} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1147,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3580802637$
$L(\frac12)$  $\approx$  $0.3580802637$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{31,\;37\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{31,\;37\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
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$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 84 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 162 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−19.6796643447, −19.1776524589, −18.3184704168, −18.0599802812, −17.7074521516, −17.1767362154, −16.5487118288, −15.7939062789, −15.4106332174, −14.8648509061, −13.9934805582, −13.5459591256, −12.5284443842, −11.92968707, −11.3794152738, −10.9296658968, −10.242014861, −9.29902572873, −8.75279519863, −7.85919901055, −7.37499164473, −6.08488341397, −5.57199266392, −4.40582703386, −2.54320858181, 2.54320858181, 4.40582703386, 5.57199266392, 6.08488341397, 7.37499164473, 7.85919901055, 8.75279519863, 9.29902572873, 10.242014861, 10.9296658968, 11.3794152738, 11.92968707, 12.5284443842, 13.5459591256, 13.9934805582, 14.8648509061, 15.4106332174, 15.7939062789, 16.5487118288, 17.1767362154, 17.7074521516, 18.0599802812, 18.3184704168, 19.1776524589, 19.6796643447

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