Properties

Label 4-6e6-1.1-c2e2-0-2
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

Downloads

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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 − 2.85·14-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 − 5.71·56-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 46656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.014281671\)
\(L(\frac12)\) \(\approx\) \(1.014281671\)
\(L(2)\) 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

−11.85671570361954823804944276768, −11.78715110271217362737285844752, −11.29005213847793890637327014065, −10.76037388878610014864115787410, −10.27978769367116423858490834813, −10.08155865108176105104635207341, −9.079673366848945864617047251226, −8.923397525866502799064398357835, −8.265620305993445971749482473171, −8.241060442128074863351176404074, −7.29507221141366825213814538117, −7.11767169636600702515750809281, −6.46050015094601654344179515554, −5.85841523828381449001317060469, −5.01327199838095338496976859364, −4.26188923778289367870301917943, −3.21886762797629263830336544099, −2.45819352788878694786123102535, −1.49416466079572913012717171047, −0.858968217759776172909653026379, 0.858968217759776172909653026379, 1.49416466079572913012717171047, 2.45819352788878694786123102535, 3.21886762797629263830336544099, 4.26188923778289367870301917943, 5.01327199838095338496976859364, 5.85841523828381449001317060469, 6.46050015094601654344179515554, 7.11767169636600702515750809281, 7.29507221141366825213814538117, 8.241060442128074863351176404074, 8.265620305993445971749482473171, 8.923397525866502799064398357835, 9.079673366848945864617047251226, 10.08155865108176105104635207341, 10.27978769367116423858490834813, 10.76037388878610014864115787410, 11.29005213847793890637327014065, 11.78715110271217362737285844752, 11.85671570361954823804944276768

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