# Properties

 Label 2-546-13.3-c1-0-5 Degree $2$ Conductor $546$ Sign $0.0128 - 0.999i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 0.999·12-s + (2.5 + 2.59i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 0.999·18-s + (−2.5 + 4.33i)19-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 0.288·12-s + (0.693 + 0.720i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 0.235·18-s + (−0.573 + 0.993i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.0128 - 0.999i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (211, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.0128 - 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.01854 + 1.00556i$$ $$L(\frac12)$$ $$\approx$$ $$1.01854 + 1.00556i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
3 $$1 + (0.5 + 0.866i)T$$
7 $$1 + (0.5 - 0.866i)T$$
13 $$1 + (-2.5 - 2.59i)T$$
good5 $$1 + 5T^{2}$$
11 $$1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2}$$
17 $$1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 - 3T + 47T^{2}$$
53 $$1 + 3T + 53T^{2}$$
59 $$1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + 16T + 73T^{2}$$
79 $$1 + 13T + 79T^{2}$$
83 $$1 - 18T + 83T^{2}$$
89 $$1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-8 + 13.8i)T + (-48.5 - 84.0i)T^{2}$$
show more
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−11.27467675444554506627348569976, −10.02605263889822091944870710653, −9.122340129898367408359917653256, −8.206461721416714192763360892477, −7.20688672234468133435500442623, −6.50643086392239607624689136621, −5.62708270514104307497978292971, −4.56427042930626092281714506410, −3.36605741552866436779474637925, −1.69814671503214991208220847581, 0.826149236098139701427696371742, 2.78155526401293814983481276927, 3.82168485186125513243235329820, 4.70971873196188448076294038605, 5.93282311916951321499209389756, 6.57202235152663544342250571000, 8.218760440740126732519333153441, 8.862927880834299839115338554495, 10.17921788330878460074369981724, 10.43665446237303050423048562125