# Properties

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

# Origins of factors

## 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$C_2^2$ $$1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4}$$
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\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