Properties

Label 1-135-135.2-r0-0-0
Degree $1$
Conductor $135$
Sign $0.0281 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0281 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0281 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.0281 + 0.999i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.0281 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6465839751 + 0.6286013576i\)
\(L(\frac12)\) \(\approx\) \(0.6465839751 + 0.6286013576i\)
\(L(1)\) \(\approx\) \(0.7843434355 + 0.4429938623i\)
\(L(1)\) \(\approx\) \(0.7843434355 + 0.4429938623i\)

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

−28.076577031200062027530962254046, −27.67138504762267242881645250246, −26.45313820456515645635844578718, −25.69376705224180154924860867129, −24.25307446100330850350867451824, −22.99634661274386874200481340634, −22.359767851042419693757554444982, −20.68918495412025600279618776297, −20.61167010437365294260661274181, −19.375549612015988918416238760586, −18.0454665693668698739641753343, −17.57074452678582853126841097629, −16.30138948341427876962588678375, −14.75961263265470336781696766482, −13.63776759111201844511495421540, −12.63611339559711737832961219887, −11.508346741891468186409505052414, −10.48342409048485798253554047881, −9.62715830266033323101070078854, −8.160503219521121821387047394459, −7.31250537488351368702515845494, −5.18560342427939566233196297331, −4.07614686659135662776162647997, −2.645454293184059119413574013138, −1.09148185342715545172777845406, 1.60220040359184682775810604439, 3.78518411303727277730217927027, 5.36389818020839391230846898799, 6.10365802060042731161155589048, 7.66571452343185379834849267853, 8.518604179406631954496638865775, 9.55787224698573770787511373072, 10.93802780365636782253735865153, 12.19172713295242288877644378566, 13.78554663087240734283299779803, 14.44374340398332122917357855215, 15.67193934359933433570873968386, 16.46786851973783632256697523596, 17.58541142459657564498829274505, 18.65656972374548797366685823642, 19.20831753090720585466686785351, 20.98786947227301479950220566767, 21.831515621152833245176147547977, 23.106582806178749454963127809570, 24.0897085473471385180674821422, 24.72833691822663581422017856347, 25.847404663388427031479495793899, 26.69172784078516030782732831474, 27.72619300878607197908498906597, 28.42460180732454202116951917670

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