Properties

Label 4-456e2-1.1-c1e2-0-18
Degree $4$
Conductor $207936$
Sign $1$
Analytic cond. $13.2581$
Root an. cond. $1.90818$
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  + 2·3-s + 9-s + 10·11-s − 8·13-s + 16·23-s − 9·25-s − 4·27-s + 20·33-s + 20·37-s − 16·39-s − 2·47-s − 5·49-s + 12·59-s − 26·61-s + 32·69-s + 4·71-s + 18·73-s − 18·75-s − 11·81-s − 24·83-s − 16·97-s + 10·99-s + 4·107-s + 40·111-s − 8·117-s + 53·121-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 3.01·11-s − 2.21·13-s + 3.33·23-s − 9/5·25-s − 0.769·27-s + 3.48·33-s + 3.28·37-s − 2.56·39-s − 0.291·47-s − 5/7·49-s + 1.56·59-s − 3.32·61-s + 3.85·69-s + 0.474·71-s + 2.10·73-s − 2.07·75-s − 1.22·81-s − 2.63·83-s − 1.62·97-s + 1.00·99-s + 0.386·107-s + 3.79·111-s − 0.739·117-s + 4.81·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(207936\)    =    \(2^{6} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(13.2581\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 207936,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.847565459\)
\(L(\frac12)\) \(\approx\) \(2.847565459\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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

−9.084561409834522310688102275122, −8.880140284265606450558573740236, −8.064901679875409643106905234825, −7.62429229152375892340957919668, −7.28807747269950922759271760406, −6.64017519777900431546557110668, −6.44323514092267885981017350010, −5.61259928782815339019051631078, −4.96316680572620019147167415239, −4.17517686433721218560152257029, −4.15537276955970084853193543262, −3.09871623117322073739701851963, −2.82511261070703297705797963037, −1.94226086883320328992152894180, −1.10108411325297450707135792130, 1.10108411325297450707135792130, 1.94226086883320328992152894180, 2.82511261070703297705797963037, 3.09871623117322073739701851963, 4.15537276955970084853193543262, 4.17517686433721218560152257029, 4.96316680572620019147167415239, 5.61259928782815339019051631078, 6.44323514092267885981017350010, 6.64017519777900431546557110668, 7.28807747269950922759271760406, 7.62429229152375892340957919668, 8.064901679875409643106905234825, 8.880140284265606450558573740236, 9.084561409834522310688102275122

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