Properties

Degree 4
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s + 5-s + 3·6-s + 2·7-s − 8-s + 4·9-s − 10-s − 2·11-s − 3·12-s − 2·14-s − 3·15-s + 16-s + 5·17-s − 4·18-s + 4·19-s + 20-s − 6·21-s + 2·22-s + 4·23-s + 3·24-s − 4·25-s + 2·28-s − 10·29-s + 3·30-s + 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.755·7-s − 0.353·8-s + 4/3·9-s − 0.316·10-s − 0.603·11-s − 0.866·12-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 1.21·17-s − 0.942·18-s + 0.917·19-s + 0.223·20-s − 1.30·21-s + 0.426·22-s + 0.834·23-s + 0.612·24-s − 4/5·25-s + 0.377·28-s − 1.85·29-s + 0.547·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 799680 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 799680 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(799680\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{799680} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 799680,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
7$C_1$ \( ( 1 - T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−12.3052609901, −12.0583153008, −11.5122503488, −11.2313614144, −10.9107885576, −10.7251654321, −10.1299959584, −9.76554711946, −9.46085477591, −8.95868385142, −8.45501219895, −7.79244516305, −7.57571100089, −7.26205913591, −6.69424462695, −6.0171571911, −5.64026114091, −5.57928681743, −5.10819261967, −4.42980862022, −3.9124907959, −2.99504188047, −2.51295843855, −1.48060460757, −1.09876168171, 0, 1.09876168171, 1.48060460757, 2.51295843855, 2.99504188047, 3.9124907959, 4.42980862022, 5.10819261967, 5.57928681743, 5.64026114091, 6.0171571911, 6.69424462695, 7.26205913591, 7.57571100089, 7.79244516305, 8.45501219895, 8.95868385142, 9.46085477591, 9.76554711946, 10.1299959584, 10.7251654321, 10.9107885576, 11.2313614144, 11.5122503488, 12.0583153008, 12.3052609901

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