Properties

Label 1-135-135.122-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: $\chi_{135} (122, \cdot )$
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

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

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