Properties

 Label 2-189-21.20-c1-0-7 Degree $2$ Conductor $189$ Sign $-0.377 + 0.925i$ Analytic cond. $1.50917$ Root an. cond. $1.22848$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Learn more

Dirichlet series

 L(s)  = 1 − 1.41i·2-s − 1.73·5-s + (1 − 2.44i)7-s − 2.82i·8-s + 2.44i·10-s − 1.41i·11-s − 2.44i·13-s + (−3.46 − 1.41i)14-s − 4.00·16-s + 5.19·17-s + 7.34i·19-s − 2.00·22-s + 2.82i·23-s − 2.00·25-s − 3.46·26-s + ⋯
 L(s)  = 1 − 0.999i·2-s − 0.774·5-s + (0.377 − 0.925i)7-s − 0.999i·8-s + 0.774i·10-s − 0.426i·11-s − 0.679i·13-s + (−0.925 − 0.377i)14-s − 1.00·16-s + 1.26·17-s + 1.68i·19-s − 0.426·22-s + 0.589i·23-s − 0.400·25-s − 0.679·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$189$$    =    $$3^{3} \cdot 7$$ Sign: $-0.377 + 0.925i$ Analytic conductor: $$1.50917$$ Root analytic conductor: $$1.22848$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{189} (188, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 189,\ (\ :1/2),\ -0.377 + 0.925i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.658358 - 0.979882i$$ $$L(\frac12)$$ $$\approx$$ $$0.658358 - 0.979882i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (-1 + 2.44i)T$$
good2 $$1 + 1.41iT - 2T^{2}$$
5 $$1 + 1.73T + 5T^{2}$$
11 $$1 + 1.41iT - 11T^{2}$$
13 $$1 + 2.44iT - 13T^{2}$$
17 $$1 - 5.19T + 17T^{2}$$
19 $$1 - 7.34iT - 19T^{2}$$
23 $$1 - 2.82iT - 23T^{2}$$
29 $$1 - 7.07iT - 29T^{2}$$
31 $$1 + 2.44iT - 31T^{2}$$
37 $$1 - 5T + 37T^{2}$$
41 $$1 - 8.66T + 41T^{2}$$
43 $$1 - 5T + 43T^{2}$$
47 $$1 + 8.66T + 47T^{2}$$
53 $$1 - 11.3iT - 53T^{2}$$
59 $$1 - 8.66T + 59T^{2}$$
61 $$1 - 2.44iT - 61T^{2}$$
67 $$1 - 2T + 67T^{2}$$
71 $$1 + 14.1iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 13T + 79T^{2}$$
83 $$1 - 1.73T + 83T^{2}$$
89 $$1 + 10.3T + 89T^{2}$$
97 $$1 - 17.1iT - 97T^{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

−12.13141041694563760704111852812, −11.25401025099405174551874117609, −10.52010450562826378801543801606, −9.700248769112340950507436858839, −8.006987996081415846114091344901, −7.42781759225687197034043682863, −5.81826367315714878428283556666, −4.06322499893154788370364151799, −3.26072537580287945895711806105, −1.21390625503354902557172281617, 2.50285706865570168890423261532, 4.47567587387250422591413427531, 5.60233471199409282489903453001, 6.75682140315435819717536517767, 7.72969249655452372306215068052, 8.507092058131594793711533661069, 9.639994425197906743321190836037, 11.33486793844748845137857339627, 11.67120188013919176218549325931, 12.83217219130144172886084074141