# Properties

 Label 1-571-571.387-r0-0-0 Degree $1$ Conductor $571$ Sign $-0.763 - 0.645i$ Analytic cond. $2.65171$ Root an. cond. $2.65171$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + 2-s + (0.309 − 0.951i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + 8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + 16-s + (−0.809 + 0.587i)17-s + ⋯
 L(s)  = 1 + 2-s + (0.309 − 0.951i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + 8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + 16-s + (−0.809 + 0.587i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$571$$ Sign: $-0.763 - 0.645i$ Analytic conductor: $$2.65171$$ Root analytic conductor: $$2.65171$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{571} (387, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 571,\ (0:\ ),\ -0.763 - 0.645i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7204616123 - 1.969005154i$$ $$L(\frac12)$$ $$\approx$$ $$0.7204616123 - 1.969005154i$$ $$L(1)$$ $$\approx$$ $$1.368855669 - 0.9709798477i$$ $$L(1)$$ $$\approx$$ $$1.368855669 - 0.9709798477i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad571 $$1$$
good2 $$1 + T$$
3 $$1 + (0.309 - 0.951i)T$$
5 $$1 + (-0.809 - 0.587i)T$$
7 $$1 + (0.309 - 0.951i)T$$
11 $$1 + (-0.809 + 0.587i)T$$
13 $$1 + (-0.809 - 0.587i)T$$
17 $$1 + (-0.809 + 0.587i)T$$
19 $$1 + (-0.809 - 0.587i)T$$
23 $$1 + (0.309 - 0.951i)T$$
29 $$1 + T$$
31 $$1 + T$$
37 $$1 + (-0.809 - 0.587i)T$$
41 $$1 + T$$
43 $$1 + (0.309 + 0.951i)T$$
47 $$1 + T$$
53 $$1 + (-0.809 + 0.587i)T$$
59 $$1 + T$$
61 $$1 + (0.309 - 0.951i)T$$
67 $$1 + (-0.809 + 0.587i)T$$
71 $$1 + (0.309 - 0.951i)T$$
73 $$1 + (-0.809 - 0.587i)T$$
79 $$1 + (0.309 - 0.951i)T$$
83 $$1 + (0.309 - 0.951i)T$$
89 $$1 + (0.309 - 0.951i)T$$
97 $$1 + (-0.809 - 0.587i)T$$
show more
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−23.45114796562153578258052988016, −22.53215199158199507200594629608, −21.92863253873550180137699024541, −21.24933145316889284268711389668, −20.606051099528126008799889576721, −19.32274346251424402818890713624, −19.126267327340831720312902828802, −17.57695034617819062364500614905, −16.360652075675009679023823428475, −15.60361519679566240449518429326, −15.27794532891696020724898819668, −14.37248631473510799433449869064, −13.69398225897739670186618392372, −12.33918307433721620535319896240, −11.5932558436248897546817239752, −10.932851274537442735317937206220, −10.04369910915846221207845529982, −8.69884486126357371262276457599, −7.90478141536281945175013598596, −6.78058715265252167458951386468, −5.617153210346486434991461254133, −4.79139393471479661977361371865, −3.98691762300402416911207677332, −2.84395997906474959027632216362, −2.37331779961891567577857212530, 0.6986100155355982337596936362, 2.08726499001818929049024874498, 3.009303045086737281089708658828, 4.35355798181221805687727105968, 4.82947208441376306609313376397, 6.27031510897920005398344335568, 7.2182370416930783461870024918, 7.76728194610666947763782155270, 8.631198460788824261158483091739, 10.36881814985495641283759443627, 11.11293219748024950318009205106, 12.24134645455000764633348646514, 12.75669825841396362631958406828, 13.35716724347570693335422711328, 14.409609324043726445527077808682, 15.127217948619415347373703684466, 15.95142089350710006945577124104, 17.170925281632166279948168981080, 17.65031417942528504863003781884, 19.29592918585957536289516967177, 19.63822804405652768125967129139, 20.48036101283860086424562813786, 20.98423451667934742970674830910, 22.40084414305921031926863461908, 23.27734173263218106475711094698