# Properties

 Degree $2$ Conductor $1323$ Sign $-0.388 - 0.921i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.23 + 2.13i)2-s + (−2.02 − 3.51i)4-s − 2.59·5-s + 5.05·8-s + (3.19 − 5.52i)10-s − 4.51·11-s + (0.5 − 0.866i)13-s + (−2.16 + 3.74i)16-s + (0.472 − 0.819i)17-s + (−2.02 − 3.51i)19-s + (5.25 + 9.10i)20-s + (5.55 − 9.61i)22-s + 0.273·23-s + 1.72·25-s + (1.23 + 2.13i)26-s + ⋯
 L(s)  = 1 + (−0.869 + 1.50i)2-s + (−1.01 − 1.75i)4-s − 1.15·5-s + 1.78·8-s + (1.00 − 1.74i)10-s − 1.36·11-s + (0.138 − 0.240i)13-s + (−0.540 + 0.936i)16-s + (0.114 − 0.198i)17-s + (−0.465 − 0.805i)19-s + (1.17 + 2.03i)20-s + (1.18 − 2.05i)22-s + 0.0569·23-s + 0.345·25-s + (0.241 + 0.417i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1323$$    =    $$3^{3} \cdot 7^{2}$$ Sign: $-0.388 - 0.921i$ Motivic weight: $$1$$ Character: $\chi_{1323} (361, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1323,\ (\ :1/2),\ -0.388 - 0.921i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4750163994$$ $$L(\frac12)$$ $$\approx$$ $$0.4750163994$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + 2.59T + 5T^{2}$$
11 $$1 + 4.51T + 11T^{2}$$
13 $$1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-0.472 + 0.819i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 - 0.273T + 23T^{2}$$
29 $$1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (0.890 + 1.54i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-3.20 + 5.54i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (3.13 - 5.43i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-1.36 - 2.36i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.90 - 13.6i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 3.27T + 71T^{2}$$
73 $$1 + (0.753 - 1.30i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (7.35 - 12.7i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-0.472 - 0.819i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-7.17 - 12.4i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (5.74 + 9.95i)T + (-48.5 + 84.0i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−9.624195166157102691219144166236, −8.633702448768682665607583601830, −8.243287361141036072655291100617, −7.40056619496888941206540224428, −7.03992504395277884258401914365, −5.85669547831551875935625633431, −5.12673145133918021779399322459, −4.18749640783575758825557304325, −2.76401802381129087944607339259, −0.64197793722663956859630046809, 0.48234140602606020994439297457, 1.99415505762733069542259324745, 3.01022195203214727651648329254, 3.85624823435101183952641076334, 4.67038054700404674091273163295, 6.05476359209257732361962146642, 7.56095936100572915157595766154, 7.946147872800205258204570206044, 8.628974628577996208052880778050, 9.548065054802384281895162376553