Properties

Degree 4
Conductor $ 2 \cdot 3 \cdot 191 $
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 − 2·5-s − 6-s − 2·7-s − 8-s − 2·10-s + 5·11-s + 12-s − 3·13-s − 2·14-s + 2·15-s + 3·16-s + 2·17-s + 2·19-s + 2·20-s + 2·21-s + 5·22-s − 6·23-s + 24-s − 3·26-s + 4·27-s + 2·28-s − 3·29-s + 2·30-s − 2·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s − 0.632·10-s + 1.50·11-s + 0.288·12-s − 0.832·13-s − 0.534·14-s + 0.516·15-s + 3/4·16-s + 0.485·17-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 1.06·22-s − 1.25·23-s + 0.204·24-s − 0.588·26-s + 0.769·27-s + 0.377·28-s − 0.557·29-s + 0.365·30-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1146 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1146 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1146\)    =    \(2 \cdot 3 \cdot 191\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1146} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1146,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5142229726$
$L(\frac12)$  $\approx$  $0.5142229726$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;191\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;191\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
191$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
13$D_{4}$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$D_{4}$ \( 1 - 15 T + 118 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 11 T + 110 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−19.7769368831, −19.2631787133, −18.8647643669, −18.0335576706, −17.5024061982, −16.7804648021, −16.5689809828, −15.8161578976, −15.1941405594, −14.4526183116, −14.1349337236, −13.4819137518, −12.5559935775, −12.1435887222, −11.9114420556, −10.9922354479, −10.0835627734, −9.52984557942, −8.72849435480, −7.75865971961, −6.98868378494, −6.07116488273, −5.22306325661, −4.15496410091, −3.54551622492, 3.54551622492, 4.15496410091, 5.22306325661, 6.07116488273, 6.98868378494, 7.75865971961, 8.72849435480, 9.52984557942, 10.0835627734, 10.9922354479, 11.9114420556, 12.1435887222, 12.5559935775, 13.4819137518, 14.1349337236, 14.4526183116, 15.1941405594, 15.8161578976, 16.5689809828, 16.7804648021, 17.5024061982, 18.0335576706, 18.8647643669, 19.2631787133, 19.7769368831

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