# Properties

 Label 2-208-13.9-c3-0-3 Degree $2$ Conductor $208$ Sign $-0.477 - 0.878i$ Analytic cond. $12.2723$ Root an. cond. $3.50319$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 + 2.59i)3-s + 2·5-s + (−2.5 − 4.33i)7-s + (9 + 15.5i)9-s + (6.5 − 11.2i)11-s + (−13 + 45.0i)13-s + (−3 + 5.19i)15-s + (−13.5 − 23.3i)17-s + (37.5 + 64.9i)19-s + 15.0·21-s + (−93.5 + 161. i)23-s − 121·25-s − 135·27-s + (6.5 − 11.2i)29-s + 104·31-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + 0.178·5-s + (−0.134 − 0.233i)7-s + (0.333 + 0.577i)9-s + (0.178 − 0.308i)11-s + (−0.277 + 0.960i)13-s + (−0.0516 + 0.0894i)15-s + (−0.192 − 0.333i)17-s + (0.452 + 0.784i)19-s + 0.155·21-s + (−0.847 + 1.46i)23-s − 0.967·25-s − 0.962·27-s + (0.0416 − 0.0720i)29-s + 0.602·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$208$$    =    $$2^{4} \cdot 13$$ Sign: $-0.477 - 0.878i$ Analytic conductor: $$12.2723$$ Root analytic conductor: $$3.50319$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{208} (113, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 208,\ (\ :3/2),\ -0.477 - 0.878i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.608410 + 1.02326i$$ $$L(\frac12)$$ $$\approx$$ $$0.608410 + 1.02326i$$ $$L(\frac{5}{2})$$ 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 + (13 - 45.0i)T$$
good3 $$1 + (1.5 - 2.59i)T + (-13.5 - 23.3i)T^{2}$$
5 $$1 - 2T + 125T^{2}$$
7 $$1 + (2.5 + 4.33i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (-6.5 + 11.2i)T + (-665.5 - 1.15e3i)T^{2}$$
17 $$1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-37.5 - 64.9i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (93.5 - 161. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-6.5 + 11.2i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 - 104T + 2.97e4T^{2}$$
37 $$1 + (211.5 - 366. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (97.5 - 168. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-99.5 - 172. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + 388T + 1.03e5T^{2}$$
53 $$1 - 618T + 1.48e5T^{2}$$
59 $$1 + (-245.5 - 425. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (87.5 + 151. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-408.5 + 707. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (-39.5 - 68.4i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 - 230T + 3.89e5T^{2}$$
79 $$1 + 764T + 4.93e5T^{2}$$
83 $$1 - 732T + 5.71e5T^{2}$$
89 $$1 + (-520.5 + 901. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (-48.5 - 84.0i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$