# Properties

 Label 2-546-39.8-c1-0-20 Degree $2$ Conductor $546$ Sign $0.239 + 0.971i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)2-s + (1.15 − 1.29i)3-s − 1.00i·4-s + (0.616 − 0.616i)5-s + (0.0958 + 1.72i)6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.331 − 2.98i)9-s + 0.872i·10-s + (−2.63 − 2.63i)11-s + (−1.29 − 1.15i)12-s + (1.08 − 3.43i)13-s + 1.00i·14-s + (−0.0836 − 1.50i)15-s − 1.00·16-s − 5.36·17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + (0.666 − 0.745i)3-s − 0.500i·4-s + (0.275 − 0.275i)5-s + (0.0391 + 0.706i)6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.110 − 0.993i)9-s + 0.275i·10-s + (−0.793 − 0.793i)11-s + (−0.372 − 0.333i)12-s + (0.300 − 0.953i)13-s + 0.267i·14-s + (−0.0215 − 0.389i)15-s − 0.250·16-s − 1.30·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.239 + 0.971i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (281, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.239 + 0.971i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.05010 - 0.822951i$$ $$L(\frac12)$$ $$\approx$$ $$1.05010 - 0.822951i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 - 0.707i)T$$
3 $$1 + (-1.15 + 1.29i)T$$
7 $$1 + (-0.707 + 0.707i)T$$
13 $$1 + (-1.08 + 3.43i)T$$
good5 $$1 + (-0.616 + 0.616i)T - 5iT^{2}$$
11 $$1 + (2.63 + 2.63i)T + 11iT^{2}$$
17 $$1 + 5.36T + 17T^{2}$$
19 $$1 + (-3.60 - 3.60i)T + 19iT^{2}$$
23 $$1 - 2.67T + 23T^{2}$$
29 $$1 - 3.00iT - 29T^{2}$$
31 $$1 + (6.60 + 6.60i)T + 31iT^{2}$$
37 $$1 + (-4.44 + 4.44i)T - 37iT^{2}$$
41 $$1 + (-8.43 + 8.43i)T - 41iT^{2}$$
43 $$1 + 0.610iT - 43T^{2}$$
47 $$1 + (-3.46 - 3.46i)T + 47iT^{2}$$
53 $$1 - 0.464iT - 53T^{2}$$
59 $$1 + (-8.93 - 8.93i)T + 59iT^{2}$$
61 $$1 + 8.37T + 61T^{2}$$
67 $$1 + (-1.77 - 1.77i)T + 67iT^{2}$$
71 $$1 + (0.00980 - 0.00980i)T - 71iT^{2}$$
73 $$1 + (-11.1 + 11.1i)T - 73iT^{2}$$
79 $$1 - 6.11T + 79T^{2}$$
83 $$1 + (3.29 - 3.29i)T - 83iT^{2}$$
89 $$1 + (-8.90 - 8.90i)T + 89iT^{2}$$
97 $$1 + (5.41 + 5.41i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$