Properties

Degree 4
Conductor $ 3^{6} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

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

\( d \)  =  \(4\)
\( N \)  =  \(729\)    =    \(3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{27} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 729,\ (\ :1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $1.37020$
$L(\frac12)$  $\approx$  $1.37020$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 4. If $p = 3$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
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$C_2^2$ \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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