# Properties

 Label 2-810-45.29-c2-0-20 Degree $2$ Conductor $810$ Sign $0.700 - 0.713i$ Analytic cond. $22.0709$ Root an. cond. $4.69796$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (4.98 + 0.387i)5-s + (−5.04 − 2.91i)7-s + 2.82·8-s + (−4 + 5.83i)10-s + (14.2 + 8.24i)11-s + (7.14 − 4.12i)14-s + (−2.00 + 3.46i)16-s − 11.3·17-s + 12·19-s + (−4.31 − 9.02i)20-s + (−20.1 + 11.6i)22-s + (−12.0 − 20.8i)23-s + (24.6 + 3.86i)25-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.996 + 0.0775i)5-s + (−0.721 − 0.416i)7-s + 0.353·8-s + (−0.400 + 0.583i)10-s + (1.29 + 0.749i)11-s + (0.510 − 0.294i)14-s + (−0.125 + 0.216i)16-s − 0.665·17-s + 0.631·19-s + (−0.215 − 0.451i)20-s + (−0.918 + 0.530i)22-s + (−0.522 − 0.905i)23-s + (0.987 + 0.154i)25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$810$$    =    $$2 \cdot 3^{4} \cdot 5$$ Sign: $0.700 - 0.713i$ Analytic conductor: $$22.0709$$ Root analytic conductor: $$4.69796$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{810} (269, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 810,\ (\ :1),\ 0.700 - 0.713i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.797267891$$ $$L(\frac12)$$ $$\approx$$ $$1.797267891$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 - 1.22i)T$$
3 $$1$$
5 $$1 + (-4.98 - 0.387i)T$$
good7 $$1 + (5.04 + 2.91i)T + (24.5 + 42.4i)T^{2}$$
11 $$1 + (-14.2 - 8.24i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (84.5 - 146. i)T^{2}$$
17 $$1 + 11.3T + 289T^{2}$$
19 $$1 - 12T + 361T^{2}$$
23 $$1 + (12.0 + 20.8i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + (420.5 + 728. i)T^{2}$$
31 $$1 + (-16 - 27.7i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 - 23.3iT - 1.36e3T^{2}$$
41 $$1 + (-49.9 + 28.8i)T + (840.5 - 1.45e3i)T^{2}$$
43 $$1 + (-35.3 - 20.4i)T + (924.5 + 1.60e3i)T^{2}$$
47 $$1 + (-17.6 + 30.6i)T + (-1.10e3 - 1.91e3i)T^{2}$$
53 $$1 - 67.8T + 2.80e3T^{2}$$
59 $$1 + (14.2 - 8.24i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (5.04 - 2.91i)T + (2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 116. iT - 5.32e3T^{2}$$
79 $$1 + (-36 + 62.3i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (-21.9 + 37.9i)T + (-3.44e3 - 5.96e3i)T^{2}$$
89 $$1 - 65.9iT - 7.92e3T^{2}$$
97 $$1 + (-141. - 81.6i)T + (4.70e3 + 8.14e3i)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

−9.992417769577871803464233881245, −9.296317284072582824651047613014, −8.685759548308482384985137824805, −7.33033388749883999415295210898, −6.64508371396697623172426942535, −6.10797881647032857323501927395, −4.90954395648796504264687282166, −3.87628299778469141597971918994, −2.35091522434072452793597887015, −1.01401447120688504147301332463, 0.911650478742015936349948412801, 2.16902848157013755140490252538, 3.22493994575206901383580772123, 4.29852244240578629994874126854, 5.76213679151366452910928728523, 6.25659062188589362618634815092, 7.39741972286507245754135643079, 8.650552888060987218367217120102, 9.357081382949592094228609378210, 9.654610809557787210496940402256