Properties

Label 1-135-135.38-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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 + 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.766 + 0.642i)11-s + (0.984 + 0.173i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.642 − 0.766i)22-s + (0.342 + 0.939i)23-s − 26-s + i·28-s + (0.173 + 0.984i)29-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 + 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.766 + 0.642i)11-s + (0.984 + 0.173i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.642 − 0.766i)22-s + (0.342 + 0.939i)23-s − 26-s + i·28-s + (0.173 + 0.984i)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: $\chi_{135} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.0880 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4395426297 + 0.4024082369i\)
\(L(\frac12)\) \(\approx\) \(0.4395426297 + 0.4024082369i\)
\(L(1)\) \(\approx\) \(0.6156824237 + 0.2088188214i\)
\(L(1)\) \(\approx\) \(0.6156824237 + 0.2088188214i\)

Euler product

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

−28.50490270019407822735834314915, −27.15340329283612101124725179006, −26.39935034531995090610654095769, −25.75534071091911146445878720806, −24.40820696197094906392076668614, −23.578627178376707679753636780925, −22.19406542377833212967180761922, −20.931630271477821013346383463884, −20.20397800869405897582852835682, −19.17980871698828847628982530176, −18.21136119436731776969964309062, −17.220245498114237134829037213253, −16.21060596708987962711635886331, −15.42253509729946877630169955022, −13.69359578811517437887584343568, −12.76348419512883880813197260388, −11.07225372590331187376072486797, −10.669352264440217410706352656979, −9.29161265234936837715663476117, −8.23030611888327765828280393508, −7.12044928004346421093223298605, −5.976242342447282599456014709569, −3.953265311124499077906573739169, −2.58756801798366522256727347296, −0.744451520793586945313394343223, 1.74544926527504204462105847306, 3.10186859956241919907761800042, 5.26766075963253298263568153539, 6.40200927519449975580327366568, 7.6365838756912724324590518581, 8.80345535648550844690567494107, 9.6603038111235834004574572275, 10.89695675747253099997813549205, 11.93598245857296392898114755458, 13.178364405770055627751061601839, 14.82064470602313729239636507479, 15.74247294015938720900361610554, 16.44408733905070233331909682101, 18.17019008943507930825981802320, 18.21342280624432645918346355505, 19.59861686164697020325483585871, 20.57433293417489176310784375536, 21.56115081576066988612557895052, 22.96273856996135403847801801141, 24.01485423990718906967313823582, 25.25664552500898646618428402991, 25.65722147512610217699219568950, 26.84426694923762296136364263477, 27.8227804620747623006374136959, 28.70938344317854547110453152967

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