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Properties

Label 1-111-111.77-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: $\chi_{111} (77, \cdot )$
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

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

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