# Properties

 Degree $2$ Conductor $810$ Sign $0.958 + 0.285i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.86 − 1.23i)5-s + (1.73 − i)7-s + 0.999i·8-s + (−1 + 2i)10-s + (−1 − 1.73i)11-s + (5.19 + 3i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s + 2i·17-s + (−0.133 − 2.23i)20-s + (1.73 + 0.999i)22-s + (−3.46 − 2i)23-s + (1.96 − 4.59i)25-s − 6·26-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.834 − 0.550i)5-s + (0.654 − 0.377i)7-s + 0.353i·8-s + (−0.316 + 0.632i)10-s + (−0.301 − 0.522i)11-s + (1.44 + 0.832i)13-s + (−0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s + 0.485i·17-s + (−0.0299 − 0.499i)20-s + (0.369 + 0.213i)22-s + (−0.722 − 0.417i)23-s + (0.392 − 0.919i)25-s − 1.17·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$810$$    =    $$2 \cdot 3^{4} \cdot 5$$ Sign: $0.958 + 0.285i$ Motivic weight: $$1$$ Character: $\chi_{810} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 810,\ (\ :1/2),\ 0.958 + 0.285i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.46819 - 0.213728i$$ $$L(\frac12)$$ $$\approx$$ $$1.46819 - 0.213728i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1$$
5 $$1 + (-1.86 + 1.23i)T$$
good7 $$1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-5.19 - 3i)T + (6.5 + 11.2i)T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 - 12T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2}$$
89 $$1 - 10T + 89T^{2}$$
97 $$1 + (6.92 - 4i)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

−10.07043924504394797487230300394, −9.314184446680741202334492893447, −8.358491566096782196957093176088, −8.051752025428131474417149523636, −6.54772183649938157911162299386, −6.06096053503610792140618667317, −4.99256660246070083994003521149, −3.91573912978990611958041656851, −2.14009939371197766997804279024, −1.07704626064651636921553692714, 1.39407201238702464335676466093, 2.48227465803464685672432730936, 3.55550428933728432772196903035, 5.06433871285258856648427331210, 5.95173467864363145707395547817, 6.89981233279765125201687627531, 7.930758303777527837428978444792, 8.627215778749335225751369756863, 9.544502804184044147836713221632, 10.34113678340372005457924874613