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Properties

Label 1-187-187.24-r0-0-0
Degree $1$
Conductor $187$
Sign $-0.170 - 0.985i$
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  + (−0.156 − 0.987i)2-s + (−0.996 + 0.0784i)3-s + (−0.951 + 0.309i)4-s + (0.522 − 0.852i)5-s + (0.233 + 0.972i)6-s + (−0.0784 + 0.996i)7-s + (0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (0.587 − 0.809i)13-s + (0.996 − 0.0784i)14-s + (−0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.996 + 0.0784i)3-s + (−0.951 + 0.309i)4-s + (0.522 − 0.852i)5-s + (0.233 + 0.972i)6-s + (−0.0784 + 0.996i)7-s + (0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (0.587 − 0.809i)13-s + (0.996 − 0.0784i)14-s + (−0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.170 - 0.985i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ -0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4913417368 - 0.5838214919i\)
\(L(\frac12)\) \(\approx\) \(0.4913417368 - 0.5838214919i\)
\(L(1)\) \(\approx\) \(0.6462051975 - 0.4003395045i\)
\(L(1)\) \(\approx\) \(0.6462051975 - 0.4003395045i\)

Euler product

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

−27.09638922439356825975677049027, −26.531660229986404742911342615081, −25.63736182744807217324309184432, −24.431153912131972991939078869640, −23.60700207409926057831826261031, −22.79447721203721294481588806604, −22.18756171135216772050821564504, −21.0172726016938179504699782133, −19.31378373054201682029214509571, −18.24482787553763823735963824418, −17.80599151718108004792355251210, −16.582391408898985159203875157746, −16.18446328126132929991490809557, −14.65877105783815560465052441495, −13.86442613162724428726048296109, −12.90327404448206834764258585479, −11.29652009771114999485372540277, −10.35773406306379663788978708694, −9.53419394468987855106173753087, −7.79929246555736558059982924642, −6.77533756370162621916372919565, −6.24479229252955063435085573026, −4.95701290101196835089576471737, −3.74583852184056114943971581832, −1.30233204456505590948515860960, 0.88328539418074695235177101361, 2.25266900325010995936051256844, 3.94943543970267367810977778773, 5.31553172533561336437514254271, 5.80138290825267491929608994151, 7.874778093662950889150090073131, 9.18545962861198563822968950414, 9.84510797613449561590300251943, 11.13942059531417266979597152797, 11.95586258024468922128853489488, 12.81496407489551050400431876273, 13.57344602329208421643992302077, 15.388737698605460096617094271121, 16.40520667632027202667995947337, 17.58915073952162109573248461446, 18.0257093158527405253015943467, 19.12622441410864075463343487688, 20.41361883062488412674324570014, 21.201021877559699031262504163831, 22.03115372018856654979687172285, 22.738441966003086968483867735906, 23.93670940960102445252086601179, 24.915339685633245855548542434233, 26.10798175832800131953291913297, 27.5132047024050696901185499189

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