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Properties

Label 1-57-57.8-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 $0$

Origins

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Normalization:  

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: $\chi_{57} (8, \cdot )$
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

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

Graph of the $Z$-function along the critical line