Properties

Degree 4
Conductor $ 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 5-s − 6-s − 3·7-s + 8-s − 3·9-s + 10-s + 2·11-s − 12-s + 8·13-s + 3·14-s − 15-s − 16-s + 2·17-s + 3·18-s + 20-s − 3·21-s − 2·22-s − 5·23-s + 24-s − 7·25-s − 8·26-s − 4·27-s + 3·28-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 − 1.13·7-s + 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 2.21·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 1.04·23-s + 0.204·24-s − 7/5·25-s − 1.56·26-s − 0.769·27-s + 0.566·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(847\)    =    \(7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{847} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 847,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3365454249$
$L(\frac12)$  $\approx$  $0.3365454249$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−19.3371476811, −19.1857249719, −18.9379033395, −17.9414335735, −17.6248621746, −17.0336103204, −16.0831318816, −15.9140726033, −15.2627959733, −14.2904799741, −13.6611723936, −13.5686390571, −12.5838491212, −11.5331064488, −11.4512586103, −10.0355090972, −9.75732511261, −8.67891590055, −8.60353961929, −7.82867506007, −6.36261389471, −5.9644397811, −4.07930883035, −3.26888344603, 3.26888344603, 4.07930883035, 5.9644397811, 6.36261389471, 7.82867506007, 8.60353961929, 8.67891590055, 9.75732511261, 10.0355090972, 11.4512586103, 11.5331064488, 12.5838491212, 13.5686390571, 13.6611723936, 14.2904799741, 15.2627959733, 15.9140726033, 16.0831318816, 17.0336103204, 17.6248621746, 17.9414335735, 18.9379033395, 19.1857249719, 19.3371476811

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