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Properties

Label 1-135-135.14-r1-0-0
Degree $1$
Conductor $135$
Sign $0.835 + 0.549i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)11-s + (0.939 + 0.342i)13-s + (0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)22-s + (0.766 − 0.642i)23-s − 26-s − 28-s + (0.939 − 0.342i)29-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)11-s + (0.939 + 0.342i)13-s + (0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)22-s + (0.766 − 0.642i)23-s − 26-s − 28-s + (0.939 − 0.342i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(14.5077\)
Root analytic conductor: \(14.5077\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (1:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9548721263 + 0.2858699486i\)
\(L(\frac12)\) \(\approx\) \(0.9548721263 + 0.2858699486i\)
\(L(1)\) \(\approx\) \(0.7127316839 + 0.1029071235i\)
\(L(1)\) \(\approx\) \(0.7127316839 + 0.1029071235i\)

Euler product

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

−28.33661204249012676849790815526, −27.27962946561738294026756952512, −26.23702600827538400455256630077, −25.50992715468556977807376311968, −24.545667597101702388479471393, −23.261635564513546061132785594796, −21.782405463695008937068020166128, −21.282511918636575888821988980160, −19.801846256233489731963137126714, −19.17097377687621421896831329217, −18.20158442563772684161147872410, −17.16389931992037056389914119783, −15.97629881851984526163644383681, −15.38035756763137800921699898642, −13.45192715930526705087462469741, −12.5359288426652364339148391751, −11.240663381135712132385190208101, −10.42736174851905160001956648417, −8.98890145493221306968721458754, −8.45448145308312925620520145343, −6.83343602358647420590753462615, −5.82443794511962543652547948010, −3.62950177835782430898255487978, −2.52052829609167792833060119282, −0.74176939625912347015948890636, 0.92919202058952488192894806147, 2.60863882497421154505174447841, 4.42856376079284011123536478007, 6.20484471197492945942936103028, 7.013292668677295225352026551155, 8.23404971034753856348647730253, 9.48406143726392878988318369027, 10.29746934673491597939311300824, 11.43778239283361724465270741442, 12.85296760077538369357378612765, 14.137351550933973405261790466464, 15.40621936781833868659585356090, 16.268797464059958392215972582462, 17.185841989979702467296446302275, 18.269403301809470546148881128049, 19.16235656717316676291041100071, 20.24666356950519052289953843552, 20.94923022533659275331355286818, 22.836987811695648913973545447262, 23.345732807851920250750992588311, 24.75357812074389958754269211921, 25.58397668334136540709776358272, 26.37650100067812754577223200077, 27.28968409187495933417588345687, 28.41348974679675593528243946470

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