# Properties

 Label 2-13e2-13.9-c1-0-7 Degree $2$ Conductor $169$ Sign $-0.611 + 0.791i$ Analytic cond. $1.34947$ Root an. cond. $1.16166$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.12 − 1.94i)2-s + (0.277 − 0.480i)3-s + (−1.52 − 2.64i)4-s − 1.44·5-s + (−0.623 − 1.07i)6-s + (−1.02 − 1.77i)7-s − 2.35·8-s + (1.34 + 2.33i)9-s + (−1.62 + 2.81i)10-s + (1.27 − 2.21i)11-s − 1.69·12-s − 4.60·14-s + (−0.400 + 0.694i)15-s + (0.400 − 0.694i)16-s + (2.64 + 4.58i)17-s + 6.04·18-s + ⋯
 L(s)  = 1 + (0.794 − 1.37i)2-s + (0.160 − 0.277i)3-s + (−0.762 − 1.32i)4-s − 0.646·5-s + (−0.254 − 0.440i)6-s + (−0.387 − 0.670i)7-s − 0.833·8-s + (0.448 + 0.777i)9-s + (−0.513 + 0.889i)10-s + (0.385 − 0.667i)11-s − 0.488·12-s − 1.23·14-s + (−0.103 + 0.179i)15-s + (0.100 − 0.173i)16-s + (0.642 + 1.11i)17-s + 1.42·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $-0.611 + 0.791i$ Analytic conductor: $$1.34947$$ Root analytic conductor: $$1.16166$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{169} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 169,\ (\ :1/2),\ -0.611 + 0.791i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.702135 - 1.43030i$$ $$L(\frac12)$$ $$\approx$$ $$0.702135 - 1.43030i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1$$
good2 $$1 + (-1.12 + 1.94i)T + (-1 - 1.73i)T^{2}$$
3 $$1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + 1.44T + 5T^{2}$$
7 $$1 + (1.02 + 1.77i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (-1.27 + 2.21i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.92 - 5.06i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-0.945 + 1.63i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + 4.26T + 31T^{2}$$
37 $$1 + (2.67 - 4.63i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (0.637 - 1.10i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (3.06 + 5.31i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 2.95T + 47T^{2}$$
53 $$1 - 5.52T + 53T^{2}$$
59 $$1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (0.288 - 0.499i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (-2.29 - 3.97i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + 10.5T + 73T^{2}$$
79 $$1 + 15.7T + 79T^{2}$$
83 $$1 - 7.72T + 83T^{2}$$
89 $$1 + (3.30 - 5.72i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (5.96 + 10.3i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$