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Properties

Label 1-135-135.128-r0-0-0
Degree $1$
Conductor $135$
Sign $0.0880 + 0.996i$
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  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.984 − 0.173i)7-s + (−0.866 + 0.5i)8-s + (0.939 + 0.342i)11-s + (−0.642 + 0.766i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.342 + 0.939i)22-s + (−0.984 − 0.173i)23-s − 26-s + i·28-s + (0.766 − 0.642i)29-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.984 − 0.173i)7-s + (−0.866 + 0.5i)8-s + (0.939 + 0.342i)11-s + (−0.642 + 0.766i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.342 + 0.939i)22-s + (−0.984 − 0.173i)23-s − 26-s + i·28-s + (0.766 − 0.642i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0880 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0880 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.0880 + 0.996i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.0880 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130739705 + 1.035210103i\)
\(L(\frac12)\) \(\approx\) \(1.130739705 + 1.035210103i\)
\(L(1)\) \(\approx\) \(1.260335981 + 0.7129975290i\)
\(L(1)\) \(\approx\) \(1.260335981 + 0.7129975290i\)

Euler product

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

−28.34715749545109939254666614379, −27.55310168401157628362332793826, −26.685888286992965935170838887909, −24.876427640180947053167583562789, −24.3163814762332300224377522244, −23.25417094789158972440698148492, −21.91697907717707234742960240088, −21.66293608443700891077305807129, −20.07500497598700742302866014898, −19.74822969541269894406758966606, −18.20772085451242289769848294846, −17.452310446487030435371261957213, −15.72347426986438440827685197635, −14.68697510522554705600504932884, −13.919207957715772107440280408175, −12.65300449755335639658329558381, −11.64245760292467731975298895144, −10.82448123134713306619871807053, −9.53075825071941573678064258211, −8.30657482253844646189501799032, −6.60454293571238739978063216415, −5.273859804031560299475465009572, −4.269511635853472851356630283515, −2.801448044157863390030394870145, −1.39113382451783211138557702754, 2.11649010188938761718655395920, 4.00382890926104771972446692339, 4.81050929093226462076160161970, 6.24751782682730767991470577887, 7.33535230630001886466042818456, 8.39032426417176747597826571252, 9.66400572228483422913631329218, 11.52187493708819478978199156969, 12.12124269211191771399638772183, 13.71371390587314774557084418677, 14.35528151963037026483137563204, 15.31453834483454712620542897188, 16.58279534079632359790833343024, 17.378084510804628749318624625375, 18.35732789444400249188942730631, 19.982154624480932660622706705, 20.957809804801536223308978762593, 22.01803849093861322251520821368, 22.78943581778609697417816254252, 24.11553913353167290634093207572, 24.512480988007643828459586144974, 25.62690335821653415698417691571, 26.83478282939258140557217312058, 27.38015673330040136835612727960, 28.91541087472552796643023674998

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