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Properties

Label 1-111-111.50-r0-0-0
Degree $1$
Conductor $111$
Sign $-0.978 - 0.205i$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.984 − 0.173i)5-s + (0.173 − 0.984i)7-s + (−0.866 + 0.5i)8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.642 − 0.766i)17-s + (−0.342 − 0.939i)19-s + (0.642 + 0.766i)20-s + (−0.984 + 0.173i)22-s + (0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.984 − 0.173i)5-s + (0.173 − 0.984i)7-s + (−0.866 + 0.5i)8-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.642 − 0.766i)17-s + (−0.342 − 0.939i)19-s + (0.642 + 0.766i)20-s + (−0.984 + 0.173i)22-s + (0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ -0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07682554197 - 0.7412758471i\)
\(L(\frac12)\) \(\approx\) \(0.07682554197 - 0.7412758471i\)
\(L(1)\) \(\approx\) \(0.5933628935 - 0.6223606053i\)
\(L(1)\) \(\approx\) \(0.5933628935 - 0.6223606053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 - T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−30.45313217844554426758432503615, −28.70650523149992717239826493076, −27.55821207767544235044386852422, −26.8995826431742410209170985060, −25.63521367378578076103323801658, −24.835848627420621550476432958277, −23.783143416612335819537596387165, −22.881544730701616334568380022085, −22.06224298188578814143593802951, −20.81271151056017533553114007536, −19.370750805195214733243494830830, −18.29177055679938380678892173167, −17.33523767005828406021078259717, −15.91727250700928239281981444762, −15.212110348556247575902847072333, −14.52632125220714327992873494864, −12.655163746216930270874442357849, −12.27284117286484412847076494704, −10.51473941200531308036876776196, −8.832476320115198557782273387970, −7.95158208890600246856340553971, −6.85903619663836250656093670246, −5.413162666261942555182448215734, −4.33228461753248495721517249602, −2.79932319385551585272548004306, 0.660513364573150067194878580968, 2.72526555987331113088574397149, 4.085173205659116589470606303312, 4.94729016450240295177827133591, 6.91896418288239468684396239459, 8.313368767366230004411747572960, 9.596306949256131345168855728910, 11.07763636249581852393259322625, 11.48900682321890726974375572108, 12.97841967808999830322685605101, 13.82373708162957144572546028206, 15.05592736217847251855043981622, 16.333946039778221617030589269255, 17.62463222961047703908853588460, 19.040015320131027431147005763041, 19.63732161130319081897933033451, 20.64068027136650725359999588278, 21.6307373123368371909520716421, 22.856978664544275070878217155084, 23.695804713495403059625200797481, 24.35967870746863577811840781495, 26.61593185564101459667341822618, 26.85959949743377934582769682084, 28.065735221079921495574214920441, 29.1499262689220341628306573862

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