Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2} $
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 2·7-s − 4·8-s + 9-s − 6·12-s − 4·14-s + 5·16-s − 2·18-s + 2·19-s − 4·21-s + 8·24-s − 10·25-s + 4·27-s + 6·28-s − 12·29-s − 6·32-s + 3·36-s − 4·38-s + 12·41-s + 8·42-s + 16·43-s − 10·48-s + 3·49-s + 20·50-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 1/3·9-s − 1.73·12-s − 1.06·14-s + 5/4·16-s − 0.471·18-s + 0.458·19-s − 0.872·21-s + 1.63·24-s − 2·25-s + 0.769·27-s + 1.13·28-s − 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.648·38-s + 1.87·41-s + 1.23·42-s + 2.43·43-s − 1.44·48-s + 3/7·49-s + 2.82·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(636804\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{636804} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 636804,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5199737796$
$L(\frac12)$  $\approx$  $0.5199737796$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.528120383242540449329483894723, −7.64939596758113403959649783851, −7.57571100088867902110310233811, −7.37702716270177429317588175316, −6.62456122444702752048713041193, −5.99620539732271804666811351997, −5.67742069637127823610337477256, −5.57928681742950427486583645839, −4.73877014588666288120041203285, −4.04211015188718094752425642434, −3.65664660676813155980150748522, −2.52986897560816951792645905521, −2.16066001255085837327128798402, −1.30325043795776141185004987249, −0.52878800471857122474052096088, 0.52878800471857122474052096088, 1.30325043795776141185004987249, 2.16066001255085837327128798402, 2.52986897560816951792645905521, 3.65664660676813155980150748522, 4.04211015188718094752425642434, 4.73877014588666288120041203285, 5.57928681742950427486583645839, 5.67742069637127823610337477256, 5.99620539732271804666811351997, 6.62456122444702752048713041193, 7.37702716270177429317588175316, 7.57571100088867902110310233811, 7.64939596758113403959649783851, 8.528120383242540449329483894723

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