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Properties

Label 1-111-111.23-r0-0-0
Degree $1$
Conductor $111$
Sign $0.849 - 0.527i$
Analytic cond. $0.515481$
Root an. cond. $0.515481$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.5 + 0.866i)7-s − i·8-s + 10-s + 11-s + (−0.866 − 0.5i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s − i·23-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.5 + 0.866i)7-s − i·8-s + 10-s + 11-s + (−0.866 − 0.5i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s − i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.679433848 - 0.4790412145i\)
\(L(\frac12)\) \(\approx\) \(1.679433848 - 0.4790412145i\)
\(L(1)\) \(\approx\) \(1.634442590 - 0.3648137235i\)
\(L(1)\) \(\approx\) \(1.634442590 - 0.3648137235i\)

Euler product

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

−29.50732445492732932802855992773, −29.05675713406851541770371978969, −27.29183508717116298670693221953, −26.21654381121120642353367301335, −25.22079509415291971071835559617, −24.488003937499717572361341282089, −23.4181564656842606992401446411, −22.33629139881068269771922187760, −21.53560511806069788691422555896, −20.429760718742288139283284876105, −19.47508856859472728762545722779, −17.397278084325571806202920189318, −17.02545396231443631354075433891, −15.9092795585728910229416669022, −14.449766727631962949097192083698, −13.70340909418721439428609989998, −12.76703895992764454173185007195, −11.60423209245806260478767261035, −10.00818236517034286351878999255, −8.79274157560223888525715178803, −7.12599313018977199783229573669, −6.30470991268706743414012986898, −4.89561653577455829271327951865, −3.796885544244193839384639550290, −2.02888256713343717449494027660, 2.00695378719625515366629070085, 2.99686485153590288200444529335, 4.64769483312447963525925499354, 6.03817502938441125216215586004, 6.7167422148696059151707088052, 9.01709993961204549518800063156, 10.039412044447869220575978732, 11.17448103393858545262667326307, 12.45277917716385175725932307038, 13.24213308596741424777624301519, 14.62418391517676916162504263225, 15.11692656280181870825457236550, 16.73022775275502791485685190380, 18.06157721189284136428953668553, 19.20170466381271633941606670435, 20.0682592016487303740084605605, 21.54225376117293652291576229807, 22.0541727546135464850093573687, 22.76557554550140793629562453603, 24.38423263340127142903662620731, 25.00047071575346528428282577313, 26.091194528231615993489339919617, 27.65737887429245217872938988674, 28.62164514574065260557221949504, 29.53941121081478701088821088050

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