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Properties

Label 1-180-180.43-r0-0-0
Degree $1$
Conductor $180$
Sign $0.979 - 0.203i$
Analytic cond. $0.835916$
Root an. cond. $0.835916$
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  + (0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s + 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 − 0.866i)49-s + i·53-s + (−0.5 + 0.866i)59-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s + 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 − 0.866i)49-s + i·53-s + (−0.5 + 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(0.835916\)
Root analytic conductor: \(0.835916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (0:\ ),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247755886 - 0.1280349004i\)
\(L(\frac12)\) \(\approx\) \(1.247755886 - 0.1280349004i\)
\(L(1)\) \(\approx\) \(1.150986172 - 0.05959578971i\)
\(L(1)\) \(\approx\) \(1.150986172 - 0.05959578971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 + T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−27.10891496006995461662308281786, −26.73670531682748079558044569692, −25.26415296859714508518474842213, −24.4410239425095904137226965703, −23.79384013148157202758740663034, −22.35510618211326776795703072099, −21.63092835106601760956877061262, −20.75265193789697195350628721735, −19.47506883173669823489302691693, −18.736643631878831096042798773998, −17.51972048337208855081006529801, −16.76351039481201343855014409969, −15.46864140809056656798117821173, −14.53289333873309017614442598311, −13.67547030146616364717412176415, −12.20600045961437085917739417211, −11.51101748600717609656928257101, −10.317778473229664438196156174889, −8.99120349647003621593024144176, −8.15216140452578814082156319984, −6.816163457465083306024955766068, −5.557436671457912371943283984149, −4.45650180577082903029479704477, −2.946230958257883919928305688857, −1.47586334913344436097842806534, 1.310965258979160927966616342991, 2.86754378419040266770519535812, 4.45451512805452230808117089894, 5.298164261750483729179292395214, 7.053715268595070701248963009130, 7.6919560562418092639218574579, 9.18243120433944807714777578975, 10.145516313153380818033451828686, 11.38324877820371172065999807808, 12.23245635101436593321553098545, 13.5519805549268831853831644002, 14.50715165356162013150812081661, 15.35823047763477375555177677187, 16.71612653323519125823425108364, 17.56767727278390769160093184471, 18.36357177813481566268674875746, 19.86552752368321613307578594176, 20.35819810099196498787229094777, 21.512709689455368933943225761256, 22.59921242669407409566315450580, 23.38721471365444536881892806797, 24.62476246734739236479766362351, 25.12921000100150618710592635461, 26.5764899653348907553985003552, 27.22771322860579569026642120947

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