# Properties

 Label 2-684-19.11-c1-0-1 Degree $2$ Conductor $684$ Sign $0.813 - 0.582i$ Analytic cond. $5.46176$ Root an. cond. $2.33704$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 7-s + (3.5 + 6.06i)13-s + (4 − 1.73i)19-s + (2.5 + 4.33i)25-s + 11·31-s − 37-s + (−2.5 + 4.33i)43-s − 6·49-s + (6.5 + 11.2i)61-s + (−2.5 − 4.33i)67-s + (3.5 − 6.06i)73-s + (6.5 − 11.2i)79-s + (−3.5 − 6.06i)91-s + (−7 + 12.1i)97-s − 13·103-s + ⋯
 L(s)  = 1 − 0.377·7-s + (0.970 + 1.68i)13-s + (0.917 − 0.397i)19-s + (0.5 + 0.866i)25-s + 1.97·31-s − 0.164·37-s + (−0.381 + 0.660i)43-s − 0.857·49-s + (0.832 + 1.44i)61-s + (−0.305 − 0.529i)67-s + (0.409 − 0.709i)73-s + (0.731 − 1.26i)79-s + (−0.366 − 0.635i)91-s + (−0.710 + 1.23i)97-s − 1.28·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $0.813 - 0.582i$ Analytic conductor: $$5.46176$$ Root analytic conductor: $$2.33704$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{684} (505, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1/2),\ 0.813 - 0.582i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.41554 + 0.454602i$$ $$L(\frac12)$$ $$\approx$$ $$1.41554 + 0.454602i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
19 $$1 + (-4 + 1.73i)T$$
good5 $$1 + (-2.5 - 4.33i)T^{2}$$
7 $$1 + T + 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 11T + 31T^{2}$$
37 $$1 + T + 37T^{2}$$
41 $$1 + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$