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Properties

Label 1-111-111.17-r0-0-0
Degree $1$
Conductor $111$
Sign $0.877 - 0.479i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.877 - 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9350227831 - 0.2386859081i\)
\(L(\frac12)\) \(\approx\) \(0.9350227831 - 0.2386859081i\)
\(L(1)\) \(\approx\) \(0.9416184267 - 0.2491424792i\)
\(L(1)\) \(\approx\) \(0.9416184267 - 0.2491424792i\)

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

−29.48302129689141807672104718672, −28.394363193074524748283955054332, −27.22379989711012017350504901540, −26.305925988143552454820497633765, −25.58136581950353774070827857893, −24.513243624893255167960904267955, −23.56484343537019556671133154497, −22.594650252903091366276597963, −21.36242371576378378017549038467, −20.18874750537895133010487421613, −18.71583886691997101710447765959, −17.94130282369392822010759042752, −16.89717457282250076643807497185, −16.1364061393242948544972562810, −14.610220480864379318708902920712, −13.85284071120848546063988250945, −12.94659148030732960896613945170, −10.63471013609497503787385195864, −10.15248924170542699082358172405, −8.60830034912123152872194438424, −7.6049791568537501286454569312, −6.19793396804247133928522113172, −5.39957631924920266210104238397, −3.63927637013291257924647021823, −1.31186463455208164890533249377, 1.72334436473391793737586058257, 2.71533842882446744495436827572, 4.588942323917620379700656860649, 5.80053307573569575573612793227, 7.67564413557202772655235432061, 9.187135637560199625380293166115, 9.63905194557725802018643253889, 11.11523396290180664904755673182, 12.16700314684838845411784314741, 13.22064484855914789515682315876, 14.20968810047796307142874422855, 15.802796438662198085074910631474, 17.18558224407886961971087759788, 18.14483786548410324542968523108, 18.74652146030207946067254910747, 20.27659058851337113321286344294, 21.0819275085981717474471537801, 21.836391211182723410775738473458, 22.87224703941881199521797879687, 24.36476027489487741806720274353, 25.67913775117843113823322958764, 26.14455399431633140010763983291, 27.8744684198939520393607263082, 28.29566497543398759548878012465, 29.223513723447989406419452219749

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