# Properties

 Label 2-684-19.8-c2-0-1 Degree $2$ Conductor $684$ Sign $-0.429 - 0.902i$ 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 + (2.19 − 3.80i)5-s + 1.44·7-s − 15.1·11-s + (−14.8 + 8.57i)13-s + (−9.77 + 16.9i)17-s + (−3.34 + 18.7i)19-s + (2.19 + 3.80i)23-s + (2.84 + 4.93i)25-s + (29.3 − 16.9i)29-s + 5.62i·31-s + (3.18 − 5.51i)35-s + 23.7i·37-s + (32.2 + 18.6i)41-s + (−15.9 + 27.6i)43-s + (−39.0 − 67.7i)47-s + ⋯
 L(s)  = 1 + (0.439 − 0.760i)5-s + 0.207·7-s − 1.37·11-s + (−1.14 + 0.659i)13-s + (−0.574 + 0.995i)17-s + (−0.176 + 0.984i)19-s + (0.0955 + 0.165i)23-s + (0.113 + 0.197i)25-s + (1.01 − 0.583i)29-s + 0.181i·31-s + (0.0909 − 0.157i)35-s + 0.641i·37-s + (0.787 + 0.454i)41-s + (−0.371 + 0.643i)43-s + (−0.831 − 1.44i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $-0.429 - 0.902i$ Analytic conductor: $$18.6376$$ Root analytic conductor: $$4.31713$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{684} (217, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1),\ -0.429 - 0.902i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.7856714316$$ $$L(\frac12)$$ $$\approx$$ $$0.7856714316$$ $$L(2)$$ 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 + (3.34 - 18.7i)T$$
good5 $$1 + (-2.19 + 3.80i)T + (-12.5 - 21.6i)T^{2}$$
7 $$1 - 1.44T + 49T^{2}$$
11 $$1 + 15.1T + 121T^{2}$$
13 $$1 + (14.8 - 8.57i)T + (84.5 - 146. i)T^{2}$$
17 $$1 + (9.77 - 16.9i)T + (-144.5 - 250. i)T^{2}$$
23 $$1 + (-2.19 - 3.80i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + (-29.3 + 16.9i)T + (420.5 - 728. i)T^{2}$$
31 $$1 - 5.62iT - 961T^{2}$$
37 $$1 - 23.7iT - 1.36e3T^{2}$$
41 $$1 + (-32.2 - 18.6i)T + (840.5 + 1.45e3i)T^{2}$$
43 $$1 + (15.9 - 27.6i)T + (-924.5 - 1.60e3i)T^{2}$$
47 $$1 + (39.0 + 67.7i)T + (-1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (-9.55 + 5.51i)T + (1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (-78.4 - 45.2i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-57.1 - 98.9i)T + (-1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (98.3 - 56.7i)T + (2.24e3 - 3.88e3i)T^{2}$$
71 $$1 + (87.9 + 50.7i)T + (2.52e3 + 4.36e3i)T^{2}$$
73 $$1 + (-10.0 + 17.4i)T + (-2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (45.8 + 26.4i)T + (3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 29.2T + 6.88e3T^{2}$$
89 $$1 + (126. - 73.2i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + (-27 - 15.5i)T + (4.70e3 + 8.14e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$