Properties

Label 4-114e2-1.1-c1e2-0-11
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s + 4·6-s − 8·7-s + 4·8-s + 9-s + 6·12-s − 16·14-s + 5·16-s + 2·18-s + 2·19-s − 16·21-s + 8·24-s + 8·25-s − 4·27-s − 24·28-s − 12·29-s + 6·32-s + 3·36-s + 4·38-s − 32·42-s + 4·43-s + 10·48-s + 34·49-s + 16·50-s + 12·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s − 3.02·7-s + 1.41·8-s + 1/3·9-s + 1.73·12-s − 4.27·14-s + 5/4·16-s + 0.471·18-s + 0.458·19-s − 3.49·21-s + 1.63·24-s + 8/5·25-s − 0.769·27-s − 4.53·28-s − 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.648·38-s − 4.93·42-s + 0.609·43-s + 1.44·48-s + 34/7·49-s + 2.26·50-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.469853714\)
\(L(\frac12)\) \(\approx\) \(2.469853714\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
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

−13.53467691744317623608872832408, −13.25092394538342122552329217174, −13.02674724024610970578432370356, −12.78992616265785395876288403106, −11.93527617100321057580621015372, −11.60928323620627003007230293400, −10.54303899595967610980929629665, −10.28203756050024740552854529559, −9.433711982995668883810367783194, −9.275813778740862873684277615043, −8.587510014585810766821465324872, −7.56995308509356553363623823211, −6.94909488668822592649725569771, −6.73832795362808146064718524367, −5.77270130908789855328915338266, −5.51443916036930876837155240244, −4.04605938820785517167919607874, −3.64773973712878028607102946935, −3.00489611670191032575990616124, −2.52754969442324543840005105817, 2.52754969442324543840005105817, 3.00489611670191032575990616124, 3.64773973712878028607102946935, 4.04605938820785517167919607874, 5.51443916036930876837155240244, 5.77270130908789855328915338266, 6.73832795362808146064718524367, 6.94909488668822592649725569771, 7.56995308509356553363623823211, 8.587510014585810766821465324872, 9.275813778740862873684277615043, 9.433711982995668883810367783194, 10.28203756050024740552854529559, 10.54303899595967610980929629665, 11.60928323620627003007230293400, 11.93527617100321057580621015372, 12.78992616265785395876288403106, 13.02674724024610970578432370356, 13.25092394538342122552329217174, 13.53467691744317623608872832408

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