# Properties

 Degree $2$ Conductor $2646$ Sign $-0.0788 - 0.996i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 2·5-s + 0.999·8-s + (−1 − 1.73i)10-s − 11-s + (−3 − 5.19i)13-s + (−0.5 − 0.866i)16-s + (2.5 + 4.33i)17-s + (−3.5 + 6.06i)19-s + (−0.999 + 1.73i)20-s + (0.5 + 0.866i)22-s − 4·23-s − 25-s + (−3 + 5.19i)26-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.894·5-s + 0.353·8-s + (−0.316 − 0.547i)10-s − 0.301·11-s + (−0.832 − 1.44i)13-s + (−0.125 − 0.216i)16-s + (0.606 + 1.05i)17-s + (−0.802 + 1.39i)19-s + (−0.223 + 0.387i)20-s + (0.106 + 0.184i)22-s − 0.834·23-s − 0.200·25-s + (−0.588 + 1.01i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2646$$    =    $$2 \cdot 3^{3} \cdot 7^{2}$$ Sign: $-0.0788 - 0.996i$ Motivic weight: $$1$$ Character: $\chi_{2646} (667, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2646,\ (\ :1/2),\ -0.0788 - 0.996i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6395124594$$ $$L(\frac12)$$ $$\approx$$ $$0.6395124594$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 + 0.866i)T$$
3 $$1$$
7 $$1$$
good5 $$1 - 2T + 5T^{2}$$
11 $$1 + T + 11T^{2}$$
13 $$1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (8 - 13.8i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (2.5 - 4.33i)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

−9.217637976670619819793185243771, −8.168340322679005512462088057496, −7.937426045493439712861710692592, −6.80253076925883842234073445890, −5.68649778031052664928837546195, −5.43887385743767058580437677234, −4.09074320107573478716896028619, −3.25544249892602445848897696443, −2.23955877835577022395845030529, −1.43636455557231033643852398841, 0.21945234658843250227167295579, 1.88657746326511693864336009756, 2.55245853285665093412408415442, 4.11382886085275281724842610614, 4.88188944588736879478920717869, 5.66489699492468480610430555085, 6.43617715905182165911512521095, 7.13271091214475350525134201749, 7.76311535284028798622245006652, 8.810661984264330921149051843940