# Properties

 Degree $2$ Conductor $294$ Sign $0.997 - 0.0633i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−0.878 − 0.507i)5-s + 2.44i·6-s + 2.82·8-s + (1.5 − 2.59i)9-s + (1.24 − 0.717i)10-s + (5.12 + 8.87i)11-s + (−2.99 − 1.73i)12-s − 8.95i·13-s − 1.75·15-s + (−2.00 + 3.46i)16-s + (26.3 − 15.2i)17-s + (2.12 + 3.67i)18-s + (13.9 + 8.06i)19-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.175 − 0.101i)5-s + 0.408i·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.124 − 0.0717i)10-s + (0.465 + 0.806i)11-s + (−0.249 − 0.144i)12-s − 0.689i·13-s − 0.117·15-s + (−0.125 + 0.216i)16-s + (1.54 − 0.894i)17-s + (0.117 + 0.204i)18-s + (0.735 + 0.424i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$294$$    =    $$2 \cdot 3 \cdot 7^{2}$$ Sign: $0.997 - 0.0633i$ Motivic weight: $$2$$ Character: $\chi_{294} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 294,\ (\ :1),\ 0.997 - 0.0633i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.60896 + 0.0510011i$$ $$L(\frac12)$$ $$\approx$$ $$1.60896 + 0.0510011i$$ $$L(2)$$ 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 + (-1.5 + 0.866i)T$$
7 $$1$$
good5 $$1 + (0.878 + 0.507i)T + (12.5 + 21.6i)T^{2}$$
11 $$1 + (-5.12 - 8.87i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + 8.95iT - 169T^{2}$$
17 $$1 + (-26.3 + 15.2i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (-13.9 - 8.06i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (-3.36 + 5.82i)T + (-264.5 - 458. i)T^{2}$$
29 $$1 - 30T + 841T^{2}$$
31 $$1 + (-43.4 + 25.0i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + (15.4 - 26.7i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 + 7.10iT - 1.68e3T^{2}$$
43 $$1 + 74.4T + 1.84e3T^{2}$$
47 $$1 + (-50.4 - 29.1i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (-35.4 - 61.4i)T + (-1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (0.426 - 0.246i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (2.48 + 1.43i)T + (1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (13.5 + 23.4i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 50.6T + 5.04e3T^{2}$$
73 $$1 + (61.1 - 35.3i)T + (2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (66.9 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 104. iT - 6.88e3T^{2}$$
89 $$1 + (125. + 72.4i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + 100. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$