# Properties

 Label 2-546-91.83-c1-0-4 Degree $2$ Conductor $546$ Sign $0.985 - 0.169i$ 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 − i·3-s − 1.00i·4-s + (0.461 + 0.461i)5-s + (0.707 + 0.707i)6-s + (−2.62 − 0.292i)7-s + (0.707 + 0.707i)8-s − 9-s − 0.652·10-s + (1.60 + 1.60i)11-s − 1.00·12-s + (2.51 + 2.58i)13-s + (2.06 − 1.65i)14-s + (0.461 − 0.461i)15-s − 1.00·16-s + 6.40·17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (0.206 + 0.206i)5-s + (0.288 + 0.288i)6-s + (−0.993 − 0.110i)7-s + (0.250 + 0.250i)8-s − 0.333·9-s − 0.206·10-s + (0.483 + 0.483i)11-s − 0.288·12-s + (0.697 + 0.716i)13-s + (0.552 − 0.441i)14-s + (0.119 − 0.119i)15-s − 0.250·16-s + 1.55·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.985 - 0.169i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (265, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.985 - 0.169i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.12188 + 0.0956139i$$ $$L(\frac12)$$ $$\approx$$ $$1.12188 + 0.0956139i$$ $$L(\frac{3}{2})$$ 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 + iT$$
7 $$1 + (2.62 + 0.292i)T$$
13 $$1 + (-2.51 - 2.58i)T$$
good5 $$1 + (-0.461 - 0.461i)T + 5iT^{2}$$
11 $$1 + (-1.60 - 1.60i)T + 11iT^{2}$$
17 $$1 - 6.40T + 17T^{2}$$
19 $$1 + (-2.52 - 2.52i)T + 19iT^{2}$$
23 $$1 + 8.51iT - 23T^{2}$$
29 $$1 - 7.91T + 29T^{2}$$
31 $$1 + (-0.922 - 0.922i)T + 31iT^{2}$$
37 $$1 + (0.953 + 0.953i)T + 37iT^{2}$$
41 $$1 + (3.89 + 3.89i)T + 41iT^{2}$$
43 $$1 - 4.17iT - 43T^{2}$$
47 $$1 + (-1.41 + 1.41i)T - 47iT^{2}$$
53 $$1 - 9.10T + 53T^{2}$$
59 $$1 + (-1.17 + 1.17i)T - 59iT^{2}$$
61 $$1 - 14.4iT - 61T^{2}$$
67 $$1 + (-3.47 + 3.47i)T - 67iT^{2}$$
71 $$1 + (8.29 - 8.29i)T - 71iT^{2}$$
73 $$1 + (5.89 - 5.89i)T - 73iT^{2}$$
79 $$1 + 9.18T + 79T^{2}$$
83 $$1 + (-9.98 - 9.98i)T + 83iT^{2}$$
89 $$1 + (-8.54 + 8.54i)T - 89iT^{2}$$
97 $$1 + (-0.272 - 0.272i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$