↑

Properties

Label 1-180-180.83-r1-0-0
Degree $1$
Conductor $180$
Sign $-0.203 - 0.979i$
Analytic cond. $19.3436$
Root an. cond. $19.3436$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(19.3436\)
Root analytic conductor: \(19.3436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (1:\ ),\ -0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9006669006 - 1.106641064i\)
\(L(\frac12)\) \(\approx\) \(0.9006669006 - 1.106641064i\)
\(L(1)\) \(\approx\) \(1.010790551 - 0.2737246289i\)
\(L(1)\) \(\approx\) \(1.010790551 - 0.2737246289i\)

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 \( 1 + (-0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.35683813954171122286217254260, −26.492434423058304327286088023169, −25.351995929642073989211850807390, −24.495328617736650787692882220654, −23.651038894563982499864447748942, −22.48898276773055330482976924558, −21.59705856270205191939876283307, −20.60392747700878114179527632915, −19.764175381796734491305925776330, −18.31829285787821018927474380879, −17.89442783423936185028542648423, −16.59316137997632959976685335935, −15.49668162270375096959486751573, −14.58456490393366034526274376519, −13.62224652781007050881606876307, −12.18027903430409729668977247958, −11.60686678281853963679572235641, −10.16325888302479171092950233369, −9.220224315613296730856052044533, −7.89592606215189026727288419830, −7.02654773537234667675825013145, −5.35446857772245803746936011455, −4.62633258911190886862462560219, −2.8372197718154871843243899694, −1.60249238936812848546327147555, 0.50148724868778361089533804833, 2.13158904376378230782267103679, 3.64987480891261785769312469577, 4.94354599307733323877023634399, 6.03786185760250115965966217980, 7.625475449253205099727552588828, 8.22815007227280790683460192350, 9.77478146048393783120085759732, 10.76454623515592978962069986520, 11.72753265036905284349983345971, 12.9728222025342865316515668354, 14.01698639316297835878574604543, 14.89308444793230890591391376820, 16.071095501200249336001012987162, 17.14506327185375522182410866227, 17.94973295642371671106926222397, 19.08528242933293927873030376358, 20.11699453596262552588582340466, 21.02837771299467661557511834128, 21.95685169567166409034096681322, 23.00616853222670069929531618310, 24.331929669219647578378812551078, 24.42139208083827265304677338950, 26.156723292829080647021808508459, 26.7130349167915842375329659431

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