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Properties

Label 1-135-135.119-r1-0-0
Degree $1$
Conductor $135$
Sign $-0.893 - 0.448i$
Analytic cond. $14.5077$
Root an. cond. $14.5077$
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

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

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.893 - 0.448i$
Analytic conductor: \(14.5077\)
Root analytic conductor: \(14.5077\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (1:\ ),\ -0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2101888349 + 0.8868564293i\)
\(L(\frac12)\) \(\approx\) \(-0.2101888349 + 0.8868564293i\)
\(L(1)\) \(\approx\) \(0.6678116554 + 0.6142364193i\)
\(L(1)\) \(\approx\) \(0.6678116554 + 0.6142364193i\)

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

−27.52663741001804607418955633718, −27.26789293205957041810308864248, −25.99422531176010872455948198906, −24.46242481852666982852678234134, −23.642485790622785706901582920724, −22.58102866327103856002051055409, −21.58809896251906265594040990316, −20.59586944537554385634948212272, −20.01074275356912921302716535575, −18.50529219209006200561264878496, −18.012229051216556873737409573653, −16.684606107785965557657959904854, −15.13676462296792274675032163974, −14.09553315526133910136618523453, −13.173488947215486153442081599945, −12.023330552525429613917773364333, −10.91124030953845565439144023785, −10.222557428661515144728914375697, −8.69270239975679666126512514969, −7.740601173917938558958699850864, −5.69294258444595523610583137464, −4.6833944911562110247632564209, −3.29932132752741057908755378522, −1.9251205696162059411531272595, −0.327800050070924462192240186037, 2.07268449464039256344868467593, 4.15272796883351891026736191343, 5.059981017373159896017883320620, 6.353302022373513476977720072673, 7.58446868565009057850979990274, 8.515711377148340618855192175548, 9.68227761192870606928232060610, 11.22524696548081203542889872966, 12.53121193440522949073461739186, 13.6005957546467692075572795023, 14.750795439853280835830088769109, 15.402482002152522938286267939256, 16.62900344638584303929097720506, 17.696304377577618738330481179, 18.3409461508751963946256596067, 19.72386019600777433017638503747, 21.30473699904219401124950242954, 21.79515548624023726420276782417, 23.27836176065825857059613306716, 23.93020470106040825285266974375, 24.770038107548334091574838236047, 25.98299577944020340625220283719, 26.55531688886598277277951588743, 27.85792944981001485696165180795, 28.49758699910065832113547096646

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