Properties

Degree $4$
Conductor $142884$
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  − 2·2-s + 3·4-s − 7-s − 4·8-s − 6·11-s + 2·14-s + 5·16-s + 12·22-s − 12·23-s − 25-s − 3·28-s + 12·29-s − 6·32-s + 4·37-s − 20·43-s − 18·44-s + 24·46-s − 6·49-s + 2·50-s + 18·53-s + 4·56-s − 24·58-s + 7·64-s + 28·67-s − 8·74-s + 6·77-s + 16·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s − 1.80·11-s + 0.534·14-s + 5/4·16-s + 2.55·22-s − 2.50·23-s − 1/5·25-s − 0.566·28-s + 2.22·29-s − 1.06·32-s + 0.657·37-s − 3.04·43-s − 2.71·44-s + 3.53·46-s − 6/7·49-s + 0.282·50-s + 2.47·53-s + 0.534·56-s − 3.15·58-s + 7/8·64-s + 3.42·67-s − 0.929·74-s + 0.683·77-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(142884\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{142884} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 142884,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4733068702\)
\(L(\frac12)\) \(\approx\) \(0.4733068702\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.533580822054688698380792321620, −8.613282561215628799056639387984, −8.268048909391662018895990050852, −8.073998889221963720660118720673, −7.69509009195539198448319901638, −6.73860900674537002760989556093, −6.69870395284666594287906434089, −5.97811569986712198414996682793, −5.37363015537505503550836847886, −4.86220558498840854064591879984, −3.91930425992199330786644833836, −3.20132525085319240177530071259, −2.44850978333263328730546301919, −1.98881454498950312538243430831, −0.55647773141554817914458641913, 0.55647773141554817914458641913, 1.98881454498950312538243430831, 2.44850978333263328730546301919, 3.20132525085319240177530071259, 3.91930425992199330786644833836, 4.86220558498840854064591879984, 5.37363015537505503550836847886, 5.97811569986712198414996682793, 6.69870395284666594287906434089, 6.73860900674537002760989556093, 7.69509009195539198448319901638, 8.073998889221963720660118720673, 8.268048909391662018895990050852, 8.613282561215628799056639387984, 9.533580822054688698380792321620

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