# Properties

 Degree 2 Conductor $3^{3}$ Sign $0.939 - 0.342i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (1.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−3 + 1.73i)5-s + (−1 + 1.73i)7-s − 8.66i·8-s − 6·10-s + (1.5 + 0.866i)11-s + (2 + 3.46i)13-s + (−3 + 1.73i)14-s + (5.5 − 9.52i)16-s + 15.5i·17-s + 11·19-s + (2.99 + 1.73i)20-s + (1.5 + 2.59i)22-s + (24 − 13.8i)23-s + ⋯
 L(s)  = 1 + (0.750 + 0.433i)2-s + (−0.125 − 0.216i)4-s + (−0.600 + 0.346i)5-s + (−0.142 + 0.247i)7-s − 1.08i·8-s − 0.600·10-s + (0.136 + 0.0787i)11-s + (0.153 + 0.266i)13-s + (−0.214 + 0.123i)14-s + (0.343 − 0.595i)16-s + 0.916i·17-s + 0.578·19-s + (0.149 + 0.0866i)20-s + (0.0681 + 0.118i)22-s + (1.04 − 0.602i)23-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27$$    =    $$3^{3}$$ $$\varepsilon$$ = $0.939 - 0.342i$ motivic weight = $$2$$ character : $\chi_{27} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 27,\ (\ :1),\ 0.939 - 0.342i)$ $L(\frac{3}{2})$ $\approx$ $1.15277 + 0.203265i$ $L(\frac12)$ $\approx$ $1.15277 + 0.203265i$ $L(2)$ 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 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1$$
good2 $$1 + (-1.5 - 0.866i)T + (2 + 3.46i)T^{2}$$
5 $$1 + (3 - 1.73i)T + (12.5 - 21.6i)T^{2}$$
7 $$1 + (1 - 1.73i)T + (-24.5 - 42.4i)T^{2}$$
11 $$1 + (-1.5 - 0.866i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (-2 - 3.46i)T + (-84.5 + 146. i)T^{2}$$
17 $$1 - 15.5iT - 289T^{2}$$
19 $$1 - 11T + 361T^{2}$$
23 $$1 + (-24 + 13.8i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + (39 + 22.5i)T + (420.5 + 728. i)T^{2}$$
31 $$1 + (16 + 27.7i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 + 34T + 1.36e3T^{2}$$
41 $$1 + (-10.5 + 6.06i)T + (840.5 - 1.45e3i)T^{2}$$
43 $$1 + (-30.5 + 52.8i)T + (-924.5 - 1.60e3i)T^{2}$$
47 $$1 + (-42 - 24.2i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 - 2.80e3T^{2}$$
59 $$1 + (43.5 - 25.1i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (28 - 48.4i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + 31.1iT - 5.04e3T^{2}$$
73 $$1 - 65T + 5.32e3T^{2}$$
79 $$1 + (19 - 32.9i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (-42 - 24.2i)T + (3.44e3 + 5.96e3i)T^{2}$$
89 $$1 - 124. iT - 7.92e3T^{2}$$
97 $$1 + (-57.5 + 99.5i)T + (-4.70e3 - 8.14e3i)T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

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