# Properties

 Label 2-684-19.7-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 + (2 + 3.46i)5-s − 3·7-s − 4·11-s + (−2.5 + 4.33i)13-s + (−4 − 1.73i)19-s + (2 − 3.46i)23-s + (−5.49 + 9.52i)25-s + (4 − 6.92i)29-s + 31-s + (−6 − 10.3i)35-s − 5·37-s + (4 + 6.92i)41-s + (2.5 + 4.33i)43-s + (−4 + 6.92i)47-s + 2·49-s + ⋯
 L(s)  = 1 + (0.894 + 1.54i)5-s − 1.13·7-s − 1.20·11-s + (−0.693 + 1.20i)13-s + (−0.917 − 0.397i)19-s + (0.417 − 0.722i)23-s + (−1.09 + 1.90i)25-s + (0.742 − 1.28i)29-s + 0.179·31-s + (−1.01 − 1.75i)35-s − 0.821·37-s + (0.624 + 1.08i)41-s + (0.381 + 0.660i)43-s + (−0.583 + 1.01i)47-s + 0.285·49-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: $\chi_{684} (577, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1/2),\ -0.813 - 0.582i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.277166 + 0.863043i$$ $$L(\frac12)$$ $$\approx$$ $$0.277166 + 0.863043i$$ $$L(\frac{3}{2})$$ 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 - 3.46i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + 3T + 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-8.5 + 14.7i)T^{2}$$
23 $$1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - T + 31T^{2}$$
37 $$1 + 5T + 37T^{2}$$
41 $$1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (-8 - 13.8i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$