↑

Properties

Label 1-135-135.59-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

Related objects

Downloads

Learn more

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: $\chi_{135} (59, \cdot )$
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 \( 1 + (-0.766 - 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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