Properties

Label 1-111-111.104-r1-0-0
Degree $1$
Conductor $111$
Sign $-0.165 + 0.986i$
Analytic cond. $11.9286$
Root an. cond. $11.9286$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.5 + 0.866i)14-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.173 + 0.984i)20-s + (0.939 + 0.342i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.5 + 0.866i)14-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.173 + 0.984i)20-s + (0.939 + 0.342i)22-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ -0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4332023122 + 0.5119508288i\)
\(L(\frac12)\) \(\approx\) \(0.4332023122 + 0.5119508288i\)
\(L(1)\) \(\approx\) \(0.9610184238 - 0.1410954777i\)
\(L(1)\) \(\approx\) \(0.9610184238 - 0.1410954777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
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

−29.28718751048357307280333018054, −27.57527774413190202082959386770, −26.8485236753557029396426447477, −25.6492901935239943078261671901, −24.704735587114902500405048222098, −23.73541501560799822539094802338, −22.79736107120101108104269429803, −22.141621539024135196477241927295, −20.6153315330054134099970039315, −19.80423720601965483071958310902, −18.49749360263072615166979681334, −16.774029364155873458459809832597, −16.31125977792587148712068369038, −15.24925549592557297825715556976, −14.102683716613335650796260390606, −12.87952082311047550195330654673, −12.13998173991526431593326861758, −10.790706464354140638164299485679, −8.96059799142483669735999122832, −7.84160728158353674475092373033, −6.73628928780369318369490399535, −5.46431874829372495591315701585, −3.99972481283005711547567462324, −3.12478703579470729847771269494, −0.209941474266250669712946984092, 2.017122597865345573863561093463, 3.52926260187960309220975810618, 4.40588420411080755137566293988, 6.15042713510853149314937428580, 7.15479975400667601036246650569, 9.07894225085123568358913635024, 10.20662176543016258959828261238, 11.521352297475513434828530220213, 12.27319618132886316516128314018, 13.3266128078087492200885969908, 14.80802146879516786652329905699, 15.36648196394103525905893738937, 16.68183447163073616659639758099, 18.47103137024735441419602661468, 19.48098326039906436975445304769, 19.87823596905128171444944679648, 21.480203451193165815119689932689, 22.25023805190512440372657678053, 23.25336913761690283733942428228, 23.91094154040399070873903527607, 25.3524169740314835980912165036, 26.43695174987885711680228162105, 27.909036634600571781992608832390, 28.378046809170889315564684530303, 29.69746857800011108129553288898

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