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Properties

Label 1-185-185.39-r1-0-0
Degree $1$
Conductor $185$
Sign $0.987 - 0.159i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (−0.984 − 0.173i)2-s + (0.173 + 0.984i)3-s + (0.939 + 0.342i)4-s − i·6-s + (−0.766 + 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)12-s + (0.342 − 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)18-s + (0.984 − 0.173i)19-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.173 + 0.984i)3-s + (0.939 + 0.342i)4-s − i·6-s + (−0.766 + 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)12-s + (0.342 − 0.939i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)18-s + (0.984 − 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9983000731 - 0.08010461408i\)
\(L(\frac12)\) \(\approx\) \(0.9983000731 - 0.08010461408i\)
\(L(1)\) \(\approx\) \(0.7170161493 + 0.09218033660i\)
\(L(1)\) \(\approx\) \(0.7170161493 + 0.09218033660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.642 - 0.766i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
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

−26.74084302996235720092094668388, −25.9558922456031422948659989623, −25.35501707114782641044544911137, −24.25139369786593215696615238565, −23.53463786661433116726446461447, −22.45555099751963107986126528941, −20.74771774177410423074793441212, −19.91620228331447993063173007607, −19.21696505918492211074419701621, −18.37692264274093651785864817634, −17.2791035740145091335551466914, −16.69901112367614650857698009934, −15.35484476145480292831397505125, −14.24745250513989487857893268314, −13.09572477498996631295678606039, −12.01420325476338845528774654757, −10.990570636186416429301076124411, −9.63778835021021965198643195358, −8.860481776199137187094743962, −7.44006195310484388408496547300, −6.92668834423336722317331598805, −5.87397754800595073209224735018, −3.67346717564424898155272478187, −2.10594765101764596577310966410, −1.01187126474180426702832764919, 0.593868802032572105461937197234, 2.75384590376399029034864421267, 3.428642406116385219250952621882, 5.35193930458138546408266482545, 6.47162235415215170871005221197, 8.015831337141483118265850908522, 9.08858740319016864860634443105, 9.59675949048971318045098972403, 10.84969470316775192174614086000, 11.58427518030632362207967126888, 12.960357262098702349507087510536, 14.45492990305227954762334450863, 15.76214427367203931861944927834, 16.03106435823196591095830572008, 17.15986702468231669735545365849, 18.298504541085059276037096860041, 19.29938456533702870478041915186, 20.154676393835211553944378328638, 21.0112593740097791392571968065, 22.04628607117220190335807369354, 22.75965114761338696511490510023, 24.687192987733759755147202381354, 25.17295534584615120846617034143, 26.285810299952752431319118759587, 26.9558508484775953586501675206

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