# Properties

 Degree $2$ Conductor $261$ Sign $-0.999 - 0.0316i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.448 + 0.713i)2-s + (1.42 − 2.96i)4-s + (−4.21 − 0.962i)5-s + (−10.1 + 4.88i)7-s + (6.10 − 0.688i)8-s + (−1.20 − 3.44i)10-s + (−1.54 − 0.173i)11-s + (−11.3 + 9.02i)13-s + (−8.03 − 5.04i)14-s + (−4.97 − 6.23i)16-s + (−16.8 + 16.8i)17-s + (0.448 − 0.157i)19-s + (−8.87 + 11.1i)20-s + (−0.567 − 1.17i)22-s + (−1.55 − 6.79i)23-s + ⋯
 L(s)  = 1 + (0.224 + 0.356i)2-s + (0.356 − 0.740i)4-s + (−0.843 − 0.192i)5-s + (−1.44 + 0.697i)7-s + (0.763 − 0.0860i)8-s + (−0.120 − 0.344i)10-s + (−0.140 − 0.0157i)11-s + (−0.870 + 0.694i)13-s + (−0.574 − 0.360i)14-s + (−0.310 − 0.389i)16-s + (−0.990 + 0.990i)17-s + (0.0236 − 0.00826i)19-s + (−0.443 + 0.556i)20-s + (−0.0258 − 0.0535i)22-s + (−0.0674 − 0.295i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0316i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0316i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$261$$    =    $$3^{2} \cdot 29$$ Sign: $-0.999 - 0.0316i$ Motivic weight: $$2$$ Character: $\chi_{261} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 261,\ (\ :1),\ -0.999 - 0.0316i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.00163347 + 0.103258i$$ $$L(\frac12)$$ $$\approx$$ $$0.00163347 + 0.103258i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
29 $$1 + (-20.2 - 20.7i)T$$
good2 $$1 + (-0.448 - 0.713i)T + (-1.73 + 3.60i)T^{2}$$
5 $$1 + (4.21 + 0.962i)T + (22.5 + 10.8i)T^{2}$$
7 $$1 + (10.1 - 4.88i)T + (30.5 - 38.3i)T^{2}$$
11 $$1 + (1.54 + 0.173i)T + (117. + 26.9i)T^{2}$$
13 $$1 + (11.3 - 9.02i)T + (37.6 - 164. i)T^{2}$$
17 $$1 + (16.8 - 16.8i)T - 289iT^{2}$$
19 $$1 + (-0.448 + 0.157i)T + (282. - 225. i)T^{2}$$
23 $$1 + (1.55 + 6.79i)T + (-476. + 229. i)T^{2}$$
31 $$1 + (7.88 + 12.5i)T + (-416. + 865. i)T^{2}$$
37 $$1 + (-26.2 + 2.96i)T + (1.33e3 - 304. i)T^{2}$$
41 $$1 + (46.1 + 46.1i)T + 1.68e3iT^{2}$$
43 $$1 + (53.7 + 33.7i)T + (802. + 1.66e3i)T^{2}$$
47 $$1 + (5.72 - 50.7i)T + (-2.15e3 - 491. i)T^{2}$$
53 $$1 + (-9.09 + 39.8i)T + (-2.53e3 - 1.21e3i)T^{2}$$
59 $$1 + 23.8T + 3.48e3T^{2}$$
61 $$1 + (-23.8 + 68.2i)T + (-2.90e3 - 2.32e3i)T^{2}$$
67 $$1 + (-89.8 - 71.6i)T + (998. + 4.37e3i)T^{2}$$
71 $$1 + (-54.7 + 43.6i)T + (1.12e3 - 4.91e3i)T^{2}$$
73 $$1 + (43.5 - 69.3i)T + (-2.31e3 - 4.80e3i)T^{2}$$
79 $$1 + (-12.9 - 114. i)T + (-6.08e3 + 1.38e3i)T^{2}$$
83 $$1 + (-1.11 - 0.537i)T + (4.29e3 + 5.38e3i)T^{2}$$
89 $$1 + (-35.1 - 55.9i)T + (-3.43e3 + 7.13e3i)T^{2}$$
97 $$1 + (26.0 + 74.3i)T + (-7.35e3 + 5.86e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$