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Properties

Label 1-185-185.177-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.445 - 0.895i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + (0.866 − 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + (0.866 − 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.445 - 0.895i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9789189280 - 1.580128039i\)
\(L(\frac12)\) \(\approx\) \(0.9789189280 - 1.580128039i\)
\(L(1)\) \(\approx\) \(1.244843165 - 1.067515875i\)
\(L(1)\) \(\approx\) \(1.244843165 - 1.067515875i\)

Euler product

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

−27.24002706851563241960031309350, −26.40320831451598683338161856068, −25.53268760336526235480500937896, −24.72509244632189739556137660287, −23.98885418281135559917686020703, −22.69658634141103870549711656327, −21.83321047652519443952577567616, −20.8434032823929374332785160038, −20.28559653481183700791115332423, −18.41417261307470359312786983866, −17.99282789355383481705151776770, −16.35901611918680344928505514591, −15.63502688814891619909194815260, −14.90101458964432346335920909901, −13.906132758135817458306469840932, −13.15037863660928769675079087592, −11.80419744736158744105903724012, −10.37317224304099141349666611154, −9.04128119058995280301419162841, −8.12704862996856406526059626771, −7.46065128122478620416527967149, −5.626873861421631373770853341770, −4.87377591721765931581243544644, −3.555294764779145966307384586796, −2.3948014608166888594869521221, 1.37702292046064082481346975683, 2.390243149484393832758367814426, 3.727620298034571912588723487315, 4.74760274604398868426591121516, 6.29968615295010413444159125669, 7.74993781003121456535718417623, 8.73708891501853800609818246140, 9.95234302902164318847379401335, 11.046847400466301330481338575998, 12.06711112734727752248389092595, 13.29419457464546222180068014335, 13.81926277387929319159200746348, 14.756230678210437775790298871118, 15.80673810770498627657994163298, 17.72489002154767775449533320017, 18.35919552563850330061980502888, 19.42659993175372243060487157908, 20.25600613513886802022429323346, 21.02346926987138864408964482848, 21.75554479397829841750838957884, 23.37660249746522361841366872220, 23.86406038333446324979556762763, 24.683504655628030120443355124615, 26.293427685793170292980705994957, 26.67743098552544155973229271080

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