# Properties

 Label 2-111-111.101-c0-0-0 Degree $2$ Conductor $111$ Sign $0.957 + 0.289i$ Analytic cond. $0.0553962$ Root an. cond. $0.235364$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 1.73i·31-s − 0.999·36-s + 37-s + (1.5 + 0.866i)39-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 1.73i·31-s − 0.999·36-s + 37-s + (1.5 + 0.866i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$111$$    =    $$3 \cdot 37$$ Sign: $0.957 + 0.289i$ Analytic conductor: $$0.0553962$$ Root analytic conductor: $$0.235364$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{111} (101, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 111,\ (\ :0),\ 0.957 + 0.289i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5278752583$$ $$L(\frac12)$$ $$\approx$$ $$0.5278752583$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.5 + 0.866i)T$$
37 $$1 - T$$
good2 $$1 + (-0.5 - 0.866i)T^{2}$$
5 $$1 + (-0.5 + 0.866i)T^{2}$$
7 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
17 $$1 + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (0.5 - 0.866i)T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + 1.73iT - T^{2}$$
41 $$1 + (0.5 - 0.866i)T^{2}$$
43 $$1 - 1.73iT - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (0.5 + 0.866i)T^{2}$$
59 $$1 + (-0.5 - 0.866i)T^{2}$$
61 $$1 + (0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (0.5 - 0.866i)T^{2}$$
73 $$1 - T + T^{2}$$
79 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
83 $$1 + (0.5 + 0.866i)T^{2}$$
89 $$1 + (-0.5 - 0.866i)T^{2}$$
97 $$1 - 1.73iT - 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

−13.55016410918758895210459647472, −12.72661832174701726040763085903, −11.91496355477986837445357335560, −11.02828773337518437619578361241, −9.678190572919910280677778371520, −7.997035025629676928244448689468, −7.19538498646848317933120534875, −6.35795379858912165737410810501, −4.41656836049261226463945647116, −2.51158045381864168173649537375, 2.87499744189534508364905084632, 5.00533503922753385933189223635, 5.76006380869294265534927873326, 7.05344265564969174407287336327, 8.971057301257502278056761614457, 9.896913020367610485625389816851, 10.63744028703307335460013897015, 11.79772524475894385983329721976, 12.61525216977882003906043806501, 14.36833704050682452832322795010