Properties

Label 1-111-111.62-r1-0-0
Degree $1$
Conductor $111$
Sign $-0.850 - 0.526i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.939 + 0.342i)20-s + (−0.766 − 0.642i)22-s + (−0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (−0.173 − 0.984i)19-s + (−0.939 + 0.342i)20-s + (−0.766 − 0.642i)22-s + (−0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ -0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5456011472 - 1.919354655i\)
\(L(\frac12)\) \(\approx\) \(0.5456011472 - 1.919354655i\)
\(L(1)\) \(\approx\) \(0.9285665861 - 0.9344373591i\)
\(L(1)\) \(\approx\) \(0.9285665861 - 0.9344373591i\)

Euler product

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

−29.89585602043379156774479048109, −28.3166916929756092395004572533, −27.45407082645259629389611159581, −26.32869024667799349744035532615, −25.28447058849175479006041240968, −24.86658820818458413688075528754, −23.484794710734514766208408624772, −22.49477768632714784392595393707, −21.691028162630939663300111614698, −20.56803081485282559015564231293, −18.69003916621496390194569339232, −17.94627583337036141763866157155, −17.23874540385486348220917121881, −15.71576872345548240955743405660, −14.84424847147796771066142277902, −13.99398589426781151129329513322, −12.87284297592872490895675452522, −11.41245496654935982668355967981, −9.86933403806298517575999899205, −8.79751194302037289932349180878, −7.52295606774409013843532541453, −6.282982972287919539150920193725, −5.36967192071794529042085605623, −3.85785762330755023242120404130, −1.92968763949086467834074663928, 0.85937350998462199145991152375, 2.01936294543120655680018913502, 3.853943131523131664188479882470, 4.93566563269065894334567592885, 6.29973789996960627010483485732, 8.45905630039749074588044123656, 9.168174137673316779751216106457, 10.68628218169079344266033074458, 11.34856367551779719146465818874, 12.878213204255203743164576866990, 13.66090724735524247905809157572, 14.52116651142546674839847570326, 16.40535789208215731981931842992, 17.50026080863498708179125246390, 18.32733639470003826357051050718, 19.75397297167366823596581337462, 20.53842303568975475014338717588, 21.44738147048482467474115909071, 22.22606323747432795500013586003, 23.80427074815376905445236158601, 24.272823110569902022538900690275, 25.934386402117295533818981953463, 26.97160285196529073167364031278, 28.03130157206175529112932804635, 28.79057118129963744647974924021

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