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Properties

Label 1-187-187.61-r0-0-0
Degree $1$
Conductor $187$
Sign $0.771 - 0.636i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
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  + (−0.987 − 0.156i)2-s + (−0.760 + 0.649i)3-s + (0.951 + 0.309i)4-s + (−0.972 − 0.233i)5-s + (0.852 − 0.522i)6-s + (−0.649 + 0.760i)7-s + (−0.891 − 0.453i)8-s + (0.156 − 0.987i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (0.760 − 0.649i)14-s + (0.891 − 0.453i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.453 + 0.891i)19-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (−0.760 + 0.649i)3-s + (0.951 + 0.309i)4-s + (−0.972 − 0.233i)5-s + (0.852 − 0.522i)6-s + (−0.649 + 0.760i)7-s + (−0.891 − 0.453i)8-s + (0.156 − 0.987i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (−0.587 − 0.809i)13-s + (0.760 − 0.649i)14-s + (0.891 − 0.453i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.453 + 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.771 - 0.636i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.771 - 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3029205908 - 0.1088186488i\)
\(L(\frac12)\) \(\approx\) \(0.3029205908 - 0.1088186488i\)
\(L(1)\) \(\approx\) \(0.4145338689 + 0.009558860636i\)
\(L(1)\) \(\approx\) \(0.4145338689 + 0.009558860636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.987 - 0.156i)T \)
3 \( 1 + (-0.760 + 0.649i)T \)
5 \( 1 + (-0.972 - 0.233i)T \)
7 \( 1 + (-0.649 + 0.760i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.453 + 0.891i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.0784 - 0.996i)T \)
31 \( 1 + (0.852 + 0.522i)T \)
37 \( 1 + (0.996 - 0.0784i)T \)
41 \( 1 + (0.0784 - 0.996i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (-0.453 - 0.891i)T \)
61 \( 1 + (-0.522 - 0.852i)T \)
67 \( 1 - T \)
71 \( 1 + (0.233 - 0.972i)T \)
73 \( 1 + (0.0784 + 0.996i)T \)
79 \( 1 + (0.233 + 0.972i)T \)
83 \( 1 + (0.156 + 0.987i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.522 - 0.852i)T \)
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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

−27.29459924170667168688235672217, −26.45262116652559061011827866419, −25.54562677205370378365759829604, −24.22664712546890864173018333654, −23.692587717899647733063177463981, −22.844107083977950431772037149692, −21.57633925364928149984041251911, −19.91582382504095436221095287978, −19.45134190025758789427088733449, −18.6670014158529870015219119671, −17.540703623338564180711054438094, −16.68900139574942893833214073182, −16.00866716720419711846347657647, −14.81368518529759749231615050834, −13.30447214550308105769091069323, −12.07250169620815682199253738211, −11.280652140671308457298616337484, −10.4360350345532717461313276658, −9.14338634468944390783297891906, −7.67023962143183546923406538285, −7.12364306626393166329908656375, −6.22854117286433843370321269904, −4.51005311831426615169760927123, −2.75191677567752906465500858648, −1.00027034968787332282970553050, 0.49256554098791508795186363538, 2.741987283379357875251946288681, 4.010948106762431639924553976794, 5.56753171947756485999199678377, 6.687788263121618651760597837678, 8.00886490316367341280108306878, 9.036491532180115548030666619810, 10.08563744972077964524742198741, 10.95206298228806860883316683922, 12.20198543874098890720956227497, 12.43414228995656194743528573519, 15.03987969295289030809032086240, 15.55015779029210134334908952129, 16.46653311610110056403566301391, 17.217552022488677416847866787413, 18.448342445551802448650588150470, 19.218355566812313633732486297294, 20.27004289462650587352013370480, 21.16450367102697146091697384678, 22.31929826262074039425400776708, 23.10336511970858004968928347032, 24.391587558742538405066796919800, 25.26338684564864832714000338034, 26.55332274241053141256911828783, 27.16352372810660264007458425284

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