# Properties

 Label 1-57-57.50-r0-0-0 Degree $1$ Conductor $57$ Sign $0.910 + 0.412i$ Analytic cond. $0.264706$ Root an. cond. $0.264706$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 7-s + 8-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 20-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 7-s + 8-s + (0.5 + 0.866i)10-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 20-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$57$$    =    $$3 \cdot 19$$ Sign: $0.910 + 0.412i$ Analytic conductor: $$0.264706$$ Root analytic conductor: $$0.264706$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{57} (50, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 57,\ (0:\ ),\ 0.910 + 0.412i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7483109484 + 0.1617302909i$$ $$L(\frac12)$$ $$\approx$$ $$0.7483109484 + 0.1617302909i$$ $$L(1)$$ $$\approx$$ $$0.8513281883 + 0.1799048043i$$ $$L(1)$$ $$\approx$$ $$0.8513281883 + 0.1799048043i$$

## Euler product

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

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

−32.90306048568547712075421198640, −31.24739337289857112745292027696, −30.446152617879390507793763286465, −29.59496797127996103628688321307, −28.38564873452360956908559307786, −27.328699100989358333143617695036, −26.280619503772491880650635432604, −25.34732146789503248957021638186, −23.56772457443820980547086784067, −22.276127624035921144082573637209, −21.223975546692812711364383433978, −20.39490485067608709220241234543, −18.71594395159039120258222777219, −18.11384331703568241530198484463, −17.02135503751817426048143196246, −15.12849262722177832220009008308, −13.79178155841372511338138268617, −12.52724916337834096460782721974, −10.842624002482313633201113224687, −10.452375249730450660964972533891, −8.63543220164148258258236520899, −7.44868173570553678447793925405, −5.335857304238585495696898649244, −3.363859934917423841765885193505, −1.8984651445601071662844561530, 1.58226221395048726164092611801, 4.71053206938510178931238081736, 5.662253031671826826222342132022, 7.47570991561203363641345390717, 8.61206285760619911991365218706, 9.74824949850873798813504306780, 11.323471212912388896178555637479, 13.24229408398333811770840275641, 14.24169070637822467613608538414, 15.68424581957397016731750197336, 16.7296024897932529525132937968, 17.78760563483812715309705065872, 18.76552962218156648521964176194, 20.45580553928255699297816489735, 21.3942281478435270700072298837, 23.29478389074274021748333117821, 24.09380322549205225521300790602, 25.06259220464060502261626071073, 26.13763428195023997457914402466, 27.381797914854225049587132086892, 28.295132045765132505938839466592, 29.29225871354894843537677407119, 31.160905267074466813411283594925, 32.00925553622283187324467064870, 33.42595473636099920159709133302