Properties

Degree 4
Conductor $ 2^{5} \cdot 3 \cdot 7 $
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 − 2·5-s + 6-s + 7-s − 8-s + 2·10-s − 4·11-s − 12-s − 6·13-s − 14-s + 2·15-s + 16-s + 8·17-s + 6·19-s − 2·20-s − 21-s + 4·22-s + 8·23-s + 24-s − 6·25-s + 6·26-s + 4·27-s + 28-s − 2·30-s − 12·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.94·17-s + 1.37·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s − 6/5·25-s + 1.17·26-s + 0.769·27-s + 0.188·28-s − 0.365·30-s − 2.15·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{672} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 672,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2783373254$
$L(\frac12)$  $\approx$  $0.2783373254$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−19.8937036145, −19.5800270198, −19.0038467807, −18.2120541317, −18.0808264901, −17.2128532162, −16.5594222959, −16.3370810014, −15.5766594288, −14.7395675207, −14.6077730657, −13.4092975284, −12.322158634, −12.3052609901, −11.2313614144, −11.0205906871, −9.76554711946, −9.65679775515, −8.17328234983, −7.57571100089, −7.26958585489, −5.57928681743, −5.01357758336, −3.1482606823, 3.1482606823, 5.01357758336, 5.57928681743, 7.26958585489, 7.57571100089, 8.17328234983, 9.65679775515, 9.76554711946, 11.0205906871, 11.2313614144, 12.3052609901, 12.322158634, 13.4092975284, 14.6077730657, 14.7395675207, 15.5766594288, 16.3370810014, 16.5594222959, 17.2128532162, 18.0808264901, 18.2120541317, 19.0038467807, 19.5800270198, 19.8937036145

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