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Properties

Label 1-111-111.65-r1-0-0
Degree $1$
Conductor $111$
Sign $-0.815 + 0.578i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(11.9286\)
Root analytic conductor: \(11.9286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2698769776 + 0.8467912457i\)
\(L(\frac12)\) \(\approx\) \(0.2698769776 + 0.8467912457i\)
\(L(1)\) \(\approx\) \(0.6240277544 + 0.3642371653i\)
\(L(1)\) \(\approx\) \(0.6240277544 + 0.3642371653i\)

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

−28.70435628366880771528906352766, −27.669386464582531352960745342947, −27.06550037473566039573668891346, −25.75992901778151716077668696904, −24.911373369799539845947058521501, −23.94170178632866645722205954136, −22.51605226860155783789192728488, −21.151134839663397990427075451872, −20.11307816122791109701151770798, −19.87741330803771012698265302627, −18.16158591051684372196923762654, −17.226453424357007318727790701340, −16.62090825658322941020258540815, −15.33738267760296683351708214709, −13.71173788129549109010415216738, −12.473813854972104761329746361976, −11.57337633450569915005299802064, −10.00706948829658781373595254493, −9.46309545661383802326816529610, −7.92285037977889203369394677170, −7.15050931672952024185188568395, −5.20843784378363680407401537556, −3.677292265018810778430605767180, −1.781671820548126922493498753832, −0.51064605410426087451439460056, 1.76543208662083935922442268491, 3.12965902923393396936339430130, 5.5384822141968851708562213521, 6.49056792669199788043307843139, 7.72647004437652843483906515007, 8.95314620799474818286168730811, 9.990529866059550377042543292977, 11.1835237142268879064196156759, 12.10647882272051811092364643030, 14.28030779168198541193751746144, 14.77184479668906985492197843094, 16.10143408933098675206805824422, 17.08817168583966161148977483422, 18.48198705659656452704869935210, 18.73307770991735951688143554832, 19.95389022578972436203250337052, 21.48391455484140763556908307819, 22.22853552704069819353742983501, 23.80384544000217323291633781264, 24.71483253710383322849715110045, 25.64299255259199225336761616045, 26.617671826570705225224260495770, 27.35675362439551843105416270615, 28.51376440383082199881199077487, 29.417267962701530802246394047979

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