# Properties

 Label 2-208-13.12-c5-0-13 Degree $2$ Conductor $208$ Sign $0.832 - 0.554i$ Analytic cond. $33.3598$ Root an. cond. $5.77579$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4·3-s + 68i·5-s − 82i·7-s − 227·9-s − 390i·11-s + (507 − 338i)13-s − 272i·15-s + 1.73e3·17-s + 1.07e3i·19-s + 328i·21-s − 2.10e3·23-s − 1.49e3·25-s + 1.88e3·27-s − 1.69e3·29-s + 1.43e3i·31-s + ⋯
 L(s)  = 1 − 0.256·3-s + 1.21i·5-s − 0.632i·7-s − 0.934·9-s − 0.971i·11-s + (0.832 − 0.554i)13-s − 0.312i·15-s + 1.45·17-s + 0.682i·19-s + 0.162i·21-s − 0.829·23-s − 0.479·25-s + 0.496·27-s − 0.373·29-s + 0.267i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$208$$    =    $$2^{4} \cdot 13$$ Sign: $0.832 - 0.554i$ Analytic conductor: $$33.3598$$ Root analytic conductor: $$5.77579$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{208} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 208,\ (\ :5/2),\ 0.832 - 0.554i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.660867438$$ $$L(\frac12)$$ $$\approx$$ $$1.660867438$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$1 + (-507 + 338i)T$$
good3 $$1 + 4T + 243T^{2}$$
5 $$1 - 68iT - 3.12e3T^{2}$$
7 $$1 + 82iT - 1.68e4T^{2}$$
11 $$1 + 390iT - 1.61e5T^{2}$$
17 $$1 - 1.73e3T + 1.41e6T^{2}$$
19 $$1 - 1.07e3iT - 2.47e6T^{2}$$
23 $$1 + 2.10e3T + 6.43e6T^{2}$$
29 $$1 + 1.69e3T + 2.05e7T^{2}$$
31 $$1 - 1.43e3iT - 2.86e7T^{2}$$
37 $$1 - 8.85e3iT - 6.93e7T^{2}$$
41 $$1 - 6.76e3iT - 1.15e8T^{2}$$
43 $$1 - 1.69e4T + 1.47e8T^{2}$$
47 $$1 - 2.51e4iT - 2.29e8T^{2}$$
53 $$1 - 3.82e4T + 4.18e8T^{2}$$
59 $$1 + 2.12e4iT - 7.14e8T^{2}$$
61 $$1 + 5.45e3T + 8.44e8T^{2}$$
67 $$1 + 4.45e4iT - 1.35e9T^{2}$$
71 $$1 - 1.77e4iT - 1.80e9T^{2}$$
73 $$1 - 3.10e4iT - 2.07e9T^{2}$$
79 $$1 - 4.53e4T + 3.07e9T^{2}$$
83 $$1 - 1.24e5iT - 3.93e9T^{2}$$
89 $$1 + 1.87e4iT - 5.58e9T^{2}$$
97 $$1 + 1.21e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$