Properties

Label 4-6e6-1.1-c3e2-0-3
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·7-s − 52·11-s + 26·13-s − 188·17-s − 74·19-s − 148·23-s − 58·25-s − 288·29-s − 248·31-s + 24·35-s + 342·37-s − 256·43-s − 132·47-s + 61·49-s − 952·53-s + 208·55-s + 1.00e3·59-s − 34·61-s − 104·65-s − 866·67-s − 776·71-s + 1.87e3·73-s + 312·77-s + 182·79-s + 1.33e3·83-s + 752·85-s + ⋯
L(s)  = 1  − 0.357·5-s − 0.323·7-s − 1.42·11-s + 0.554·13-s − 2.68·17-s − 0.893·19-s − 1.34·23-s − 0.463·25-s − 1.84·29-s − 1.43·31-s + 0.115·35-s + 1.51·37-s − 0.907·43-s − 0.409·47-s + 0.177·49-s − 2.46·53-s + 0.509·55-s + 2.21·59-s − 0.0713·61-s − 0.198·65-s − 1.57·67-s − 1.29·71-s + 3.00·73-s + 0.461·77-s + 0.259·79-s + 1.76·83-s + 0.959·85-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: \(2\)
Selberg data: \((4,\ 46656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 52 T + 1718 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 2 p T + 3843 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 188 T + 18482 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 74 T + 3567 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 148 T + 25310 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 288 T + 57994 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 8 p T + 63438 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 342 T + 112547 T^{2} - 342 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 46478 T^{2} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 256 T + 172518 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 132 T + 211822 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 952 T + 489050 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 1004 T + 583382 T^{2} - 1004 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 34 T + 246171 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 866 T + 742935 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 776 T + 704366 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 182 T + 872679 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1336 T + 1208918 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 876 T + 1522402 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 38 T + 431787 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.28681890177975284825942107145, −11.20005042753862983307829275165, −10.86272433452038570441576687664, −10.23750672477883073053639386695, −9.592599347390670723041476578697, −9.217849848799679557240575756936, −8.568406116398970486285166585168, −8.163597681983709554458218544042, −7.63205285410961500652380795349, −7.11499054106274539152179193302, −6.27406080109797370552182757789, −6.14003116139899207745223457900, −5.25917762805001642040019405229, −4.60455133020779375551468754608, −4.00875344980959931260724918236, −3.43051628258172812348845148223, −2.24526278147986014803521810159, −2.02675066573573264484141549007, 0, 0, 2.02675066573573264484141549007, 2.24526278147986014803521810159, 3.43051628258172812348845148223, 4.00875344980959931260724918236, 4.60455133020779375551468754608, 5.25917762805001642040019405229, 6.14003116139899207745223457900, 6.27406080109797370552182757789, 7.11499054106274539152179193302, 7.63205285410961500652380795349, 8.163597681983709554458218544042, 8.568406116398970486285166585168, 9.217849848799679557240575756936, 9.592599347390670723041476578697, 10.23750672477883073053639386695, 10.86272433452038570441576687664, 11.20005042753862983307829275165, 11.28681890177975284825942107145

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