# Properties

 Label 2-684-19.12-c2-0-1 Degree $2$ Conductor $684$ Sign $-0.980 + 0.194i$ Analytic cond. $18.6376$ Root an. cond. $4.31713$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3 + 5.19i)5-s − 5·7-s + (−16.5 − 9.52i)13-s + (3 + 5.19i)17-s + (−13 + 13.8i)19-s + (−12 + 20.7i)23-s + (−5.5 + 9.52i)25-s + (−27 − 15.5i)29-s − 29.4i·31-s + (−15 − 25.9i)35-s − 60.6i·37-s + (36 − 20.7i)41-s + (12.5 + 21.6i)43-s + (−21 + 36.3i)47-s − 24·49-s + ⋯
 L(s)  = 1 + (0.600 + 1.03i)5-s − 0.714·7-s + (−1.26 − 0.732i)13-s + (0.176 + 0.305i)17-s + (−0.684 + 0.729i)19-s + (−0.521 + 0.903i)23-s + (−0.220 + 0.381i)25-s + (−0.931 − 0.537i)29-s − 0.949i·31-s + (−0.428 − 0.742i)35-s − 1.63i·37-s + (0.878 − 0.506i)41-s + (0.290 + 0.503i)43-s + (−0.446 + 0.773i)47-s − 0.489·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $-0.980 + 0.194i$ Analytic conductor: $$18.6376$$ Root analytic conductor: $$4.31713$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{684} (145, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1),\ -0.980 + 0.194i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2542508179$$ $$L(\frac12)$$ $$\approx$$ $$0.2542508179$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
19 $$1 + (13 - 13.8i)T$$
good5 $$1 + (-3 - 5.19i)T + (-12.5 + 21.6i)T^{2}$$
7 $$1 + 5T + 49T^{2}$$
11 $$1 + 121T^{2}$$
13 $$1 + (16.5 + 9.52i)T + (84.5 + 146. i)T^{2}$$
17 $$1 + (-3 - 5.19i)T + (-144.5 + 250. i)T^{2}$$
23 $$1 + (12 - 20.7i)T + (-264.5 - 458. i)T^{2}$$
29 $$1 + (27 + 15.5i)T + (420.5 + 728. i)T^{2}$$
31 $$1 + 29.4iT - 961T^{2}$$
37 $$1 + 60.6iT - 1.36e3T^{2}$$
41 $$1 + (-36 + 20.7i)T + (840.5 - 1.45e3i)T^{2}$$
43 $$1 + (-12.5 - 21.6i)T + (-924.5 + 1.60e3i)T^{2}$$
47 $$1 + (21 - 36.3i)T + (-1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (54 + 31.1i)T + (1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (63 - 36.3i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (21.5 - 37.2i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-49.5 - 28.5i)T + (2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + (54 - 31.1i)T + (2.52e3 - 4.36e3i)T^{2}$$
73 $$1 + (5.5 + 9.52i)T + (-2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-1.5 + 0.866i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 126T + 6.88e3T^{2}$$
89 $$1 + (-9 - 5.19i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + (114 - 65.8i)T + (4.70e3 - 8.14e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$