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Properties

Label 1-111-111.107-r1-0-0
Degree $1$
Conductor $111$
Sign $0.0357 + 0.999i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0357 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0357 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.0357 + 0.999i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ 0.0357 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.205305230 + 2.127924634i\)
\(L(\frac12)\) \(\approx\) \(2.205305230 + 2.127924634i\)
\(L(1)\) \(\approx\) \(1.721962064 + 0.8336116022i\)
\(L(1)\) \(\approx\) \(1.721962064 + 0.8336116022i\)

Euler product

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

−28.96409721882090662245757904644, −28.14981237455339236904054547385, −27.3058685373487832005835098298, −25.33026937007325987817630071562, −24.7393571235575384013366212750, −23.836458891270141263353637577, −22.74374068317873121101504273724, −21.59350257876580281693765700693, −20.92945162631777076501843902313, −19.79060739208146187343020350631, −18.88313914868170356056521291184, −17.305235713973126954211772080156, −15.874873698490333974655055651925, −15.35407074108159268116477398268, −13.8056940148172888412185537595, −12.960859730196228311870540617494, −11.87172142781841571125724630282, −11.06234286686926644335669977779, −9.31547732491624901700404744519, −8.254906324695950103297594236717, −6.32938430395536586799645392963, −5.35882945063631961634708743345, −4.179258753403433446940226512693, −2.68113730769683983318134488719, −1.03696992121378990067443876086, 2.02170734978825830922879721507, 3.684415323585065918971483573117, 4.49981559080672791830767966994, 6.48663686501517084573038592038, 6.93290352488102767847028012918, 8.36736976338427092231100575063, 10.41995203846581648261855774420, 11.19149668200117717216609504572, 12.50369882473289991715960627854, 13.751391872056292350832657300604, 14.54901048298867973236798285264, 15.45464084623868033491495314962, 16.77057681058351461540634445368, 17.705963480561769833214005970284, 19.24274863469045333240854838056, 20.33105087620750128792559702248, 21.365111283924510276437357496150, 22.52245242014322064635861666958, 23.1998668959580951414683477671, 24.028919872472636883990517447641, 25.45040987729280629310166254115, 26.154198712734840171922788738043, 27.12157960411212838615505614838, 28.685254860069835553635016405510, 29.89646113615825741224615029044

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