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Properties

Label 1-135-135.83-r0-0-0
Degree $1$
Conductor $135$
Sign $-0.906 - 0.421i$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.642 − 0.766i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.984 − 0.173i)22-s + (0.642 − 0.766i)23-s − 26-s + i·28-s + (−0.939 − 0.342i)29-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.642 − 0.766i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.984 − 0.173i)22-s + (0.642 − 0.766i)23-s − 26-s + i·28-s + (−0.939 − 0.342i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.906 - 0.421i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ -0.906 - 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1973844825 - 0.8922157794i\)
\(L(\frac12)\) \(\approx\) \(0.1973844825 - 0.8922157794i\)
\(L(1)\) \(\approx\) \(0.6930336152 - 0.6818125796i\)
\(L(1)\) \(\approx\) \(0.6930336152 - 0.6818125796i\)

Euler product

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

−28.75664054124077973889998398598, −28.019689865781559014957102244529, −26.59234252982295838776396823156, −25.95306250187922847634678064493, −24.992411569285387460945149670564, −24.100307164905234973104940123156, −23.06395826673924762049109378777, −22.12424314015951311431086735798, −21.39235799887369940028887629504, −19.84477133939342334720593479749, −18.657739676482360606194081202151, −17.669324191701350277293129445, −16.65373112657646002420601903463, −15.500870911848199167675375965896, −14.95244460267215491470490294720, −13.51951731694876195436988991296, −12.6933701882470698373317556538, −11.56659159925344499044292127639, −9.61818404438130566490341660714, −8.9473758208444360367400689187, −7.41999605598337358016666724461, −6.52716861955908358973577044244, −5.27654828541576169798020858634, −4.11311403704549026633088447978, −2.50009448261397231982032689316, 0.74639189987299175817090813827, 2.71569854450107075993197536942, 3.72606707881584073486959138782, 5.113978857216060316700911665113, 6.40397856321229516234843701042, 8.07068175184698295424235594514, 9.44673838169132533348231881958, 10.43790909106912229958131826777, 11.32343869212864969363949382411, 12.68791022870275549548899598406, 13.43573944299247595101148630969, 14.42596193985375356618423860830, 15.79290128384278963935131242645, 17.03431419679198673790630428133, 18.27277191561083298973496266423, 19.24087689512029113865491264004, 20.162322397428994110671562176451, 20.9430454733194807634924337218, 22.28865920582610270246666202781, 22.78172993793198573866101628203, 23.9443682081654148021020482776, 24.98550592959109201577154202546, 26.715605565275343628204798478, 26.96209340685445623830148353307, 28.425537389916296799815788089283

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