Properties

Label 4-6e6-1.1-c2e2-0-3
Degree $4$
Conductor $46656$
Sign $1$
Analytic cond. $34.6399$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 12·4-s + 2·5-s + 10·7-s + 32·8-s + 8·10-s − 10·11-s + 40·14-s + 80·16-s + 24·20-s − 40·22-s + 25·25-s + 120·28-s − 100·29-s − 38·31-s + 192·32-s + 20·35-s + 64·40-s − 120·44-s + 49·49-s + 100·50-s − 94·53-s − 20·55-s + 320·56-s − 400·58-s + 20·59-s − 152·62-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 2/5·5-s + 10/7·7-s + 4·8-s + 4/5·10-s − 0.909·11-s + 20/7·14-s + 5·16-s + 6/5·20-s − 1.81·22-s + 25-s + 30/7·28-s − 3.44·29-s − 1.22·31-s + 6·32-s + 4/7·35-s + 8/5·40-s − 2.72·44-s + 49-s + 2·50-s − 1.77·53-s − 0.363·55-s + 40/7·56-s − 6.89·58-s + 0.338·59-s − 2.45·62-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(34.6399\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(9.299681330\)
\(L(\frac12)\) \(\approx\) \(9.299681330\)
\(L(2)\) 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$C_1$ \( ( 1 - p T )^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 10 T - 21 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T + 483 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2^2$ \( 1 + 94 T + 6027 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 + 50 T - 2829 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2$ \( ( 1 + 58 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 134 T + 11067 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 - 190 T + 26691 T^{2} - 190 p^{2} T^{3} + p^{4} T^{4} \)
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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

−12.62151192715286089363614937032, −11.83407984584653296454658481445, −11.45512249332009800597260272414, −11.00066963430168686405502360083, −10.81884399482237922896722021657, −10.27713823805249923090290550963, −9.484174955687653977700448194286, −8.843526962115495654153450304437, −7.85879263318750141785987509295, −7.75327755894529546932625397904, −7.20793388823295564740666662322, −6.57275785743757410232237913225, −5.67799324969658571577223763054, −5.52627004688533635554540833958, −4.99919492001644686267661053982, −4.41790692035140023797704053633, −3.71405751954578190198215968930, −3.00884831610442378889702866765, −2.03561600933202312448902025052, −1.67024438877746004357248344401, 1.67024438877746004357248344401, 2.03561600933202312448902025052, 3.00884831610442378889702866765, 3.71405751954578190198215968930, 4.41790692035140023797704053633, 4.99919492001644686267661053982, 5.52627004688533635554540833958, 5.67799324969658571577223763054, 6.57275785743757410232237913225, 7.20793388823295564740666662322, 7.75327755894529546932625397904, 7.85879263318750141785987509295, 8.843526962115495654153450304437, 9.484174955687653977700448194286, 10.27713823805249923090290550963, 10.81884399482237922896722021657, 11.00066963430168686405502360083, 11.45512249332009800597260272414, 11.83407984584653296454658481445, 12.62151192715286089363614937032

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