# Properties

 Label 2-13e2-13.12-c3-0-22 Degree $2$ Conductor $169$ Sign $0.832 - 0.554i$ Analytic cond. $9.97132$ Root an. cond. $3.15774$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.56i·2-s + 8.68·3-s + 5.56·4-s + 3.56i·5-s + 13.5i·6-s − 27.1i·7-s + 21.1i·8-s + 48.4·9-s − 5.56·10-s + 15.2i·11-s + 48.3·12-s + 42.4·14-s + 30.9i·15-s + 11.4·16-s − 44.5·17-s + 75.6i·18-s + ⋯
 L(s)  = 1 + 0.552i·2-s + 1.67·3-s + 0.695·4-s + 0.318i·5-s + 0.922i·6-s − 1.46i·7-s + 0.935i·8-s + 1.79·9-s − 0.175·10-s + 0.418i·11-s + 1.16·12-s + 0.810·14-s + 0.532i·15-s + 0.178·16-s − 0.635·17-s + 0.990i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $0.832 - 0.554i$ Analytic conductor: $$9.97132$$ Root analytic conductor: $$3.15774$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{169} (168, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 169,\ (\ :3/2),\ 0.832 - 0.554i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.25541 + 0.985659i$$ $$L(\frac12)$$ $$\approx$$ $$3.25541 + 0.985659i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1$$
good2 $$1 - 1.56iT - 8T^{2}$$
3 $$1 - 8.68T + 27T^{2}$$
5 $$1 - 3.56iT - 125T^{2}$$
7 $$1 + 27.1iT - 343T^{2}$$
11 $$1 - 15.2iT - 1.33e3T^{2}$$
17 $$1 + 44.5T + 4.91e3T^{2}$$
19 $$1 + 23.9iT - 6.85e3T^{2}$$
23 $$1 + 122.T + 1.21e4T^{2}$$
29 $$1 + 219.T + 2.43e4T^{2}$$
31 $$1 + 27.0iT - 2.97e4T^{2}$$
37 $$1 - 94.1iT - 5.06e4T^{2}$$
41 $$1 - 160. iT - 6.89e4T^{2}$$
43 $$1 - 151.T + 7.95e4T^{2}$$
47 $$1 - 466. iT - 1.03e5T^{2}$$
53 $$1 + 120.T + 1.48e5T^{2}$$
59 $$1 + 439. iT - 2.05e5T^{2}$$
61 $$1 + 137.T + 2.26e5T^{2}$$
67 $$1 + 512. iT - 3.00e5T^{2}$$
71 $$1 + 410. iT - 3.57e5T^{2}$$
73 $$1 + 308. iT - 3.89e5T^{2}$$
79 $$1 + 586.T + 4.93e5T^{2}$$
83 $$1 + 1.35e3iT - 5.71e5T^{2}$$
89 $$1 - 439. iT - 7.04e5T^{2}$$
97 $$1 - 1.51e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$