Properties

Label 4-12e4-1.1-c2e2-0-3
Degree $4$
Conductor $20736$
Sign $1$
Analytic cond. $15.3955$
Root an. cond. $1.98083$
Motivic weight $2$
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  + 8·7-s + 16·13-s + 32·19-s + 32·25-s − 88·31-s − 68·37-s + 80·43-s − 50·49-s + 100·61-s − 16·67-s − 32·73-s + 152·79-s + 128·91-s + 352·97-s + 56·103-s + 112·109-s − 46·121-s + 127-s + 131-s + 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 8/7·7-s + 1.23·13-s + 1.68·19-s + 1.27·25-s − 2.83·31-s − 1.83·37-s + 1.86·43-s − 1.02·49-s + 1.63·61-s − 0.238·67-s − 0.438·73-s + 1.92·79-s + 1.40·91-s + 3.62·97-s + 0.543·103-s + 1.02·109-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s + 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(15.3955\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20736,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.208201781\)
\(L(\frac12)\) \(\approx\) \(2.208201781\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 32 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 416 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 770 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1664 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 1184 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2782 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4160 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 5810 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7490 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 76 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 334 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15680 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 176 T + p^{2} T^{2} )^{2} \)
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

−12.95144612329030438916692540454, −12.78812196102597744355154839078, −11.99285281882888634036664730608, −11.55734411058502514096370369368, −10.95036025816715436989989699316, −10.87886488031190005676882756420, −10.14994928887121089115636409590, −9.366793536857037910290904970593, −8.859447563945669516645463318685, −8.602559935029808092750102300036, −7.56535336701896195563304015038, −7.56127038151087732071704044096, −6.70842282145024197873733413351, −5.92088429781282570169664953717, −5.21592030494207403613256592313, −4.94699627927993528046479845951, −3.77186660978585751289383881603, −3.35682349327906350185366723474, −2.01410154362396858144121923750, −1.11813309660071895897224950303, 1.11813309660071895897224950303, 2.01410154362396858144121923750, 3.35682349327906350185366723474, 3.77186660978585751289383881603, 4.94699627927993528046479845951, 5.21592030494207403613256592313, 5.92088429781282570169664953717, 6.70842282145024197873733413351, 7.56127038151087732071704044096, 7.56535336701896195563304015038, 8.602559935029808092750102300036, 8.859447563945669516645463318685, 9.366793536857037910290904970593, 10.14994928887121089115636409590, 10.87886488031190005676882756420, 10.95036025816715436989989699316, 11.55734411058502514096370369368, 11.99285281882888634036664730608, 12.78812196102597744355154839078, 12.95144612329030438916692540454

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