Properties

Label 4-117e2-1.1-c3e2-0-9
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $47.6544$
Root an. cond. $2.62739$
Motivic weight $3$
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  + 3·2-s + 8·4-s + 18·5-s − 2·7-s + 45·8-s + 54·10-s + 30·11-s + 65·13-s − 6·14-s + 135·16-s − 111·17-s + 46·19-s + 144·20-s + 90·22-s − 6·23-s − 7·25-s + 195·26-s − 16·28-s − 105·29-s − 200·31-s + 360·32-s − 333·34-s − 36·35-s − 17·37-s + 138·38-s + 810·40-s − 231·41-s + ⋯
L(s)  = 1  + 1.06·2-s + 4-s + 1.60·5-s − 0.107·7-s + 1.98·8-s + 1.70·10-s + 0.822·11-s + 1.38·13-s − 0.114·14-s + 2.10·16-s − 1.58·17-s + 0.555·19-s + 1.60·20-s + 0.872·22-s − 0.0543·23-s − 0.0559·25-s + 1.47·26-s − 0.107·28-s − 0.672·29-s − 1.15·31-s + 1.98·32-s − 1.67·34-s − 0.173·35-s − 0.0755·37-s + 0.589·38-s + 3.20·40-s − 0.879·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(47.6544\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.681757131\)
\(L(\frac12)\) \(\approx\) \(6.681757131\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
13$C_2$ \( 1 - 5 p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2$ \( ( 1 - 9 T + p^{3} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 111 T + 7408 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 46 T - 4743 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 6 T - 12131 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 105 T - 13364 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 100 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 17 T - 50364 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 + 231 T - 15560 T^{2} + 231 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 514 T + 184689 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 162 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 639 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 600 T + 154621 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 233 T - 172692 T^{2} + 233 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 926 T + 556713 T^{2} + 926 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 + 930 T + 506989 T^{2} + 930 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 253 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 1324 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + 810 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 498 T - 456965 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 19 p T + p^{3} T^{2} ) \)
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

−13.43210084596291737933072821190, −12.98181605368131781045299577396, −12.63580747924479550990377985281, −11.67303944268787283790913006182, −11.13650096035384196506808555589, −11.00456295744853967184249723267, −10.10679819391296597530508796590, −9.889836037190048215400670228634, −8.855199398443890316467726803264, −8.824106990181890564379696977260, −7.46009440396919559274862177859, −7.24866087342947947889409969194, −6.30277198921207241574782966663, −5.91482495112269628691916948868, −5.56085498300444515836785084161, −4.41192177725563983346359950072, −4.11761748298800714492538725836, −3.04725332939264108393957179846, −1.83443535941580837755260848413, −1.56714298362230859634875059950, 1.56714298362230859634875059950, 1.83443535941580837755260848413, 3.04725332939264108393957179846, 4.11761748298800714492538725836, 4.41192177725563983346359950072, 5.56085498300444515836785084161, 5.91482495112269628691916948868, 6.30277198921207241574782966663, 7.24866087342947947889409969194, 7.46009440396919559274862177859, 8.824106990181890564379696977260, 8.855199398443890316467726803264, 9.889836037190048215400670228634, 10.10679819391296597530508796590, 11.00456295744853967184249723267, 11.13650096035384196506808555589, 11.67303944268787283790913006182, 12.63580747924479550990377985281, 12.98181605368131781045299577396, 13.43210084596291737933072821190

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