Properties

Degree 4
Conductor $ 3 \cdot 13 \cdot 19 $
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 − 5-s + 6-s + 7-s − 3·8-s − 2·9-s + 10-s + 11-s − 12-s − 3·13-s − 14-s + 15-s + 16-s + 3·17-s + 2·18-s + 3·19-s − 20-s − 21-s − 22-s + 3·24-s + 25-s + 3·26-s + 2·27-s + 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 0.612·24-s + 1/5·25-s + 0.588·26-s + 0.384·27-s + 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 741 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 741 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(741\)    =    \(3 \cdot 13 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{741} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 741,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2930756651$
$L(\frac12)$  $\approx$  $0.2930756651$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;13,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$V_4$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$V_4$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 52 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−19.7685200594, −19.527268027, −18.7507008438, −18.0137992912, −17.857424351, −17.1033710514, −16.6463372785, −16.0488556879, −15.4120595535, −14.5458133322, −14.4410430794, −13.3052192593, −12.3082958994, −11.8599241011, −11.4730644695, −10.7412911254, −9.80212126515, −9.23275850724, −8.31765896296, −7.67791208455, −6.67784626435, −5.82678507084, −4.78588644, −3.03062161208, 3.03062161208, 4.78588644, 5.82678507084, 6.67784626435, 7.67791208455, 8.31765896296, 9.23275850724, 9.80212126515, 10.7412911254, 11.4730644695, 11.8599241011, 12.3082958994, 13.3052192593, 14.4410430794, 14.5458133322, 15.4120595535, 16.0488556879, 16.6463372785, 17.1033710514, 17.857424351, 18.0137992912, 18.7507008438, 19.527268027, 19.7685200594

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