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Properties

Label 1-135-135.92-r0-0-0
Degree $1$
Conductor $135$
Sign $0.851 - 0.524i$
Analytic cond. $0.626937$
Root an. cond. $0.626937$
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  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.342 − 0.939i)7-s + (0.866 − 0.5i)8-s + (−0.766 + 0.642i)11-s + (−0.984 − 0.173i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.642 + 0.766i)22-s + (−0.342 − 0.939i)23-s − 26-s − i·28-s + (0.173 + 0.984i)29-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.342 − 0.939i)7-s + (0.866 − 0.5i)8-s + (−0.766 + 0.642i)11-s + (−0.984 − 0.173i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.642 + 0.766i)22-s + (−0.342 − 0.939i)23-s − 26-s − i·28-s + (0.173 + 0.984i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.831543629 - 0.5185483906i\)
\(L(\frac12)\) \(\approx\) \(1.831543629 - 0.5185483906i\)
\(L(1)\) \(\approx\) \(1.730753454 - 0.3214490730i\)
\(L(1)\) \(\approx\) \(1.730753454 - 0.3214490730i\)

Euler product

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

−28.9284424531733585808503227558, −27.75702899606517229551934817652, −26.445421365245883560668268669146, −25.42697321135735033974921060701, −24.43709331384320250827446494804, −23.81960533448811390760528573562, −22.54989974729451946525293251600, −21.66267867948681085915021831373, −21.01376787199064102894223500647, −19.73494823443737082664766199724, −18.60708688822855195879740928838, −17.296335901355093284436682813103, −16.06268742773806881627473749599, −15.27363666584834326423694395979, −14.23381650896066352012169586081, −13.22189990522592922941167183461, −12.03306593165310958933679919326, −11.34222460201308609878905042776, −9.8006565156779062779636299353, −8.225828071764796365570691428412, −7.16543248480737538751199489396, −5.63143697027105819134890539810, −5.0098989492285753744384003038, −3.28659028291898825921831975625, −2.1839986108733964110728906688, 1.68716933298047169816481307557, 3.22039548580799689166141504525, 4.50574780281354728954840622038, 5.48855242228340100165010313917, 7.04324646013965128591612145097, 7.86105959464847500684918865099, 10.04130328037753396891597791481, 10.64536637411408932893284285840, 12.13079701488727641183852651200, 12.852145334420111262257314115114, 14.18639613674649664081161112592, 14.75536091928022875878521964848, 16.12388768281518178662383339232, 17.0435114963325688495705467754, 18.44381325846501534703754467935, 19.84028669997470066277382558897, 20.495833145618082443765398592525, 21.45142049650159453286628510864, 22.605131519336336498697364723146, 23.42259939380621326177571624662, 24.21061253567331096065477216126, 25.31071800191687564362961182254, 26.37800670883725132934151734212, 27.56827825593889891141213729367, 28.792483743225609816527049972545

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