Properties

Label 4-3e6-1.1-c2e2-0-1
Degree $4$
Conductor $729$
Sign $1$
Analytic cond. $0.541249$
Root an. cond. $0.857727$
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  + 3·2-s + 2·4-s − 6·5-s − 2·7-s − 3·8-s − 18·10-s + 3·11-s + 4·13-s − 6·14-s − 3·16-s + 22·19-s − 12·20-s + 9·22-s + 48·23-s − 25-s + 12·26-s − 4·28-s − 78·29-s − 32·31-s − 12·32-s + 12·35-s − 68·37-s + 66·38-s + 18·40-s + 21·41-s + 61·43-s + 6·44-s + ⋯
L(s)  = 1  + 3/2·2-s + 1/2·4-s − 6/5·5-s − 2/7·7-s − 3/8·8-s − 9/5·10-s + 3/11·11-s + 4/13·13-s − 3/7·14-s − 0.187·16-s + 1.15·19-s − 3/5·20-s + 9/22·22-s + 2.08·23-s − 0.0399·25-s + 6/13·26-s − 1/7·28-s − 2.68·29-s − 1.03·31-s − 3/8·32-s + 0.342·35-s − 1.83·37-s + 1.73·38-s + 9/20·40-s + 0.512·41-s + 1.41·43-s + 3/22·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(0.541249\)
Root analytic conductor: \(0.857727\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 729,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.370204912\)
\(L(\frac12)\) \(\approx\) \(1.370204912\)
\(L(2)\) 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)$
bad3 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
11$C_2^2$ \( 1 - 3 T + 124 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 335 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 78 T + 2869 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 21 T + 1828 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_2^2$ \( 1 + 87 T + 6004 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9110 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 65 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \)
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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

−16.97852006073033270567206013185, −16.96053097621608547095111719016, −15.81194033586924826395631381677, −15.54281834585365900676257331795, −14.90252909148791652513749596453, −14.32332847564253113049640967588, −13.63945946502705838048830076409, −13.20879476950558931481134188137, −12.41975491966578491170170802382, −12.14421051556168363781619320952, −11.06277888648967116387068588430, −10.88481014932351468344588166574, −9.251625512923039693916806155580, −9.045968939723155070863090308782, −7.51335180285504062301562843657, −7.29002998337248567308671618499, −5.76332489329024944103761021810, −5.10449220145022378540999645957, −3.93770391264367289528376419738, −3.47851947562111442583082863489, 3.47851947562111442583082863489, 3.93770391264367289528376419738, 5.10449220145022378540999645957, 5.76332489329024944103761021810, 7.29002998337248567308671618499, 7.51335180285504062301562843657, 9.045968939723155070863090308782, 9.251625512923039693916806155580, 10.88481014932351468344588166574, 11.06277888648967116387068588430, 12.14421051556168363781619320952, 12.41975491966578491170170802382, 13.20879476950558931481134188137, 13.63945946502705838048830076409, 14.32332847564253113049640967588, 14.90252909148791652513749596453, 15.54281834585365900676257331795, 15.81194033586924826395631381677, 16.96053097621608547095111719016, 16.97852006073033270567206013185

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