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Properties

Label 1-135-135.47-r0-0-0
Degree $1$
Conductor $135$
Sign $0.851 - 0.524i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7374673771 - 0.2087924717i\)
\(L(\frac12)\) \(\approx\) \(0.7374673771 - 0.2087924717i\)
\(L(1)\) \(\approx\) \(0.7552715281 - 0.1893434648i\)
\(L(1)\) \(\approx\) \(0.7552715281 - 0.1893434648i\)

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

−28.58331982832039088495348671233, −27.4511500396274612667916706035, −26.60436428992908778702114607620, −25.6297004297349728654645900917, −24.94873686963120635102694404001, −23.71995872790554955968290178100, −22.93526187909011178195522749075, −21.83561746940584204652056984812, −20.27854371326638980960457765682, −19.28609161970906580755928563176, −18.60514820583134803450562110021, −17.26092150931128763260215420451, −16.42555736162974119538840902271, −15.661071021125815410001252099025, −14.29151644686974324749051470733, −13.508285143945032157985476304145, −11.887352938578810895089597721716, −10.586048213841293994877762537983, −9.40834490865718286468911020146, −8.693658381059049338252281836908, −7.04905908011303337774291584775, −6.44462770343363570867692781597, −4.999143922274072314195846601895, −3.33093339245452285008770796743, −1.15841186575832046197112609129, 1.24778507516884035512830531253, 2.98877328064369173567542303547, 3.94810659567636512125898936266, 5.91805500014270603445159527343, 7.30354447424951051803651206248, 8.57743724688220823899086111603, 9.6432742090010376084435287463, 10.471278965183325576549035182545, 11.864332129794693854241708178614, 12.638100314981165290096922771518, 13.742013345706771856469863774439, 15.32750746273535998376465075675, 16.52382842696028974626480558403, 17.34531269297874417462433952849, 18.60441129536738389139282362096, 19.33136966910310043399746501838, 20.30259723233074834722563408003, 21.26498568610839914483397209934, 22.47782460159198196335060712288, 23.02924678646627434952560424821, 25.01131911266560931836713351101, 25.51458678260299493559668854362, 26.63998901179395850575891889520, 27.6655785255987787916742216317, 28.38560544452661835735407827014

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