Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s + 3·8-s + 9-s + 10-s − 8·11-s − 4·13-s − 16-s + 4·17-s − 18-s + 8·19-s + 20-s + 8·22-s + 8·23-s − 2·25-s + 4·26-s + 4·29-s − 8·31-s − 5·32-s − 4·34-s − 36-s − 4·37-s − 8·38-s − 3·40-s + 4·41-s + 8·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 2.41·11-s − 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.70·22-s + 1.66·23-s − 2/5·25-s + 0.784·26-s + 0.742·29-s − 1.43·31-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 0.657·37-s − 1.29·38-s − 0.474·40-s + 0.624·41-s + 1.20·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{720} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 720,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2951333844$
$L(\frac12)$  $\approx$  $0.2951333844$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−19.8937036145, −19.110462459, −19.0038467807, −18.1058537571, −18.0808264901, −17.2359534775, −16.5594222959, −15.9590260302, −15.5766594288, −14.7395675207, −14.0069258505, −13.4092975284, −12.6461787614, −12.322158634, −11.0205906871, −10.6789224512, −9.65679775515, −9.52345167581, −8.17328234983, −7.66488013442, −7.26958585489, −5.23920392625, −5.01357758336, −3.1482606823, 3.1482606823, 5.01357758336, 5.23920392625, 7.26958585489, 7.66488013442, 8.17328234983, 9.52345167581, 9.65679775515, 10.6789224512, 11.0205906871, 12.322158634, 12.6461787614, 13.4092975284, 14.0069258505, 14.7395675207, 15.5766594288, 15.9590260302, 16.5594222959, 17.2359534775, 18.0808264901, 18.1058537571, 19.0038467807, 19.110462459, 19.8937036145

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