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Properties

Label 1-185-185.137-r1-0-0
Degree $1$
Conductor $185$
Sign $0.395 - 0.918i$
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  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + 11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + 11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.395 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7052035420 - 0.4642093612i\)
\(L(\frac12)\) \(\approx\) \(0.7052035420 - 0.4642093612i\)
\(L(1)\) \(\approx\) \(0.6214160791 - 0.1187377594i\)
\(L(1)\) \(\approx\) \(0.6214160791 - 0.1187377594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + iT \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + iT \)
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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

−27.404854649243986748774641469413, −26.31960606153706914383146819982, −24.8081920996323594137367597024, −24.61688801433382664959069180405, −23.683123031924112624234420442171, −22.473764274010209207531221633605, −21.58622616707040279003146222137, −19.9888197205654925661118826616, −19.28070457556994111190194340413, −18.0611146593458212688894432755, −17.54346049165662810400770164465, −16.81828383493883169186014548708, −15.52077066343871125950076009351, −14.71560185147908708098323683654, −13.34120875806922299100455123003, −11.772422300561795320900779634333, −11.35626416169379726564286205631, −10.06441762752295753664446397344, −8.87314601207525773643025814401, −7.72531888660333487422834532774, −6.81652054041718382126247856832, −5.70550054685955316878898009680, −4.75368028699307064697842617802, −2.19421177111678872120677286443, −1.04533407972770626260106147496, 0.55067108159199813813651340771, 1.90133811405706154553520492272, 3.80854093185291048421151726248, 4.768996747730925571963811211700, 6.486828525557028559538931639258, 7.44126137909806073540910384636, 8.874731939730282854455372254578, 9.827388925055243757451121197194, 10.81203686643493618052974867827, 11.658517065969912786712934362817, 12.33446623589763557784856469298, 14.04741289780342565434779975134, 15.28500552739685631978820396642, 16.58057587891724217089271098284, 17.08125115830285133961382119755, 17.91815840892493587941073562407, 18.93685527826899263711803085070, 20.23853021448537227219981035861, 20.85459070892313330583587233200, 21.99802082020421179230462823015, 22.61338331100706844559294440156, 24.16596952574906019841634244852, 24.79408840891123981320744294528, 26.4324318109786495210172449578, 26.98799171175444242246849617693

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