Properties

Label 4-6e6-1.1-c3e2-0-2
Degree $4$
Conductor $46656$
Sign $1$
Analytic cond. $162.420$
Root an. cond. $3.56993$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 24·7-s − 2·11-s + 32·13-s + 56·17-s + 184·19-s − 92·23-s + 95·25-s + 336·29-s + 376·31-s − 192·35-s + 348·37-s + 312·41-s + 80·43-s + 228·47-s + 43·49-s − 152·53-s + 16·55-s − 680·59-s − 112·61-s − 256·65-s + 352·67-s − 1.81e3·71-s − 574·73-s − 48·77-s − 1.36e3·79-s + 782·83-s + ⋯
L(s)  = 1  − 0.715·5-s + 1.29·7-s − 0.0548·11-s + 0.682·13-s + 0.798·17-s + 2.22·19-s − 0.834·23-s + 0.759·25-s + 2.15·29-s + 2.17·31-s − 0.927·35-s + 1.54·37-s + 1.18·41-s + 0.283·43-s + 0.707·47-s + 0.125·49-s − 0.393·53-s + 0.0392·55-s − 1.50·59-s − 0.235·61-s − 0.488·65-s + 0.641·67-s − 3.03·71-s − 0.920·73-s − 0.0710·77-s − 1.93·79-s + 1.03·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(162.420\)
Root analytic conductor: \(3.56993\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.265900058\)
\(L(\frac12)\) \(\approx\) \(3.265900058\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 8 T - 31 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 24 T + 533 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + T + p^{3} T^{2} )^{2} \)
13$D_{4}$ \( 1 - 32 T - 102 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 56 T + 5858 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 184 T + 20994 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 4 p T + 21698 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 336 T + 75814 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 376 T + 87501 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 348 T + 126830 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 312 T + 132478 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 80 T + 102402 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 228 T + 201634 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 152 T + 253337 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 680 T + 355286 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 112 T + 152970 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 352 T + 289170 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1816 T + 1497518 T^{2} + 1816 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 287 T + p^{3} T^{2} )^{2} \)
79$D_{4}$ \( 1 + 1360 T + 1144350 T^{2} + 1360 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 782 T + 992327 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 240 T + 1328110 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 338 T + 1834899 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87155239491171279064759460021, −11.83835515331070642536891061116, −11.22681793610697157558217488445, −10.78393469598445035558312701337, −10.05377568913918971547139798921, −9.905074771040248132470443821665, −9.031588197240277705946068079454, −8.549719304513415609985442299902, −7.982197569423095370460507509881, −7.72568856430581117914485565018, −7.33661269288757715741243858205, −6.34620304303927934957474450570, −5.95219174120883752919284721674, −5.20682554496329626333244700609, −4.44422655077233422999691416673, −4.35880564921701000208727633144, −3.10120864927989580428109827646, −2.80275563037754282454914468864, −1.27077781699008545401896784942, −0.955214406059313780669523281826, 0.955214406059313780669523281826, 1.27077781699008545401896784942, 2.80275563037754282454914468864, 3.10120864927989580428109827646, 4.35880564921701000208727633144, 4.44422655077233422999691416673, 5.20682554496329626333244700609, 5.95219174120883752919284721674, 6.34620304303927934957474450570, 7.33661269288757715741243858205, 7.72568856430581117914485565018, 7.982197569423095370460507509881, 8.549719304513415609985442299902, 9.031588197240277705946068079454, 9.905074771040248132470443821665, 10.05377568913918971547139798921, 10.78393469598445035558312701337, 11.22681793610697157558217488445, 11.83835515331070642536891061116, 11.87155239491171279064759460021

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