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Properties

Label 1-135-135.124-r0-0-0
Degree $1$
Conductor $135$
Sign $0.448 - 0.893i$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)11-s + (−0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)22-s + (0.939 − 0.342i)23-s + 26-s − 28-s + (0.173 + 0.984i)29-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)11-s + (−0.173 + 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)22-s + (0.939 − 0.342i)23-s + 26-s − 28-s + (0.173 + 0.984i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8667607057 - 0.5346259637i\)
\(L(\frac12)\) \(\approx\) \(0.8667607057 - 0.5346259637i\)
\(L(1)\) \(\approx\) \(0.9095316534 - 0.4081925615i\)
\(L(1)\) \(\approx\) \(0.9095316534 - 0.4081925615i\)

Euler product

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

−28.31980128195320199067048807111, −27.45175419765311897846694395925, −26.89471863955312547614810087886, −25.454660156640829759725630062191, −24.97686263511852323626713055138, −23.778155298537391352107678393842, −23.049747973041138171806781252410, −21.9638397061634975037893176256, −20.6948555645724758731807240152, −19.50459155150275046829637161571, −18.33364562430024427067452903240, −17.29259654828583536965107311419, −16.807050905529099263051104431353, −15.074975827266734925957006576463, −14.795176555777931908897560515396, −13.48695427961586803288382709399, −12.30481927147214268792125686201, −10.724412296494545383344021836080, −9.65363619764267734166319346082, −8.28721353704018055407446720030, −7.54147878928004133285398693326, −6.204799545219017688568032212241, −5.004718942558198942541303064489, −3.85237012404635370802134582625, −1.439637137362731854881770283746, 1.33388610707639769851497660640, 2.71424545211776813810265989564, 4.202929700575303049598826579766, 5.29292098920741411560673569980, 7.14983766437949596789314008824, 8.67486991500951785765802070626, 9.27295920563290551683324850092, 10.9089469784418734080599719853, 11.51749753861495009064183642201, 12.57557633165578472308608889974, 13.963717601879636665802700451149, 14.62159230153428286337232887607, 16.42377089256557641472968456155, 17.414636980524901682640756307644, 18.46872059788454439762581844409, 19.27807491243263972031624883424, 20.38153691136337577841114257091, 21.418062680738370754291930469310, 21.95479254015595554322633648865, 23.30033543904187992562757179545, 24.293896486423497551978207394075, 25.52382471097389058059678380841, 26.867983608719940510582231283154, 27.39875965790145367764512634428, 28.40689338652596953482201591288

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