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Properties

Label 1-187-187.29-r0-0-0
Degree $1$
Conductor $187$
Sign $0.800 + 0.598i$
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  + (0.891 + 0.453i)2-s + (−0.852 + 0.522i)3-s + (0.587 + 0.809i)4-s + (0.649 − 0.760i)5-s + (−0.996 + 0.0784i)6-s + (0.522 − 0.852i)7-s + (0.156 + 0.987i)8-s + (0.453 − 0.891i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (0.951 − 0.309i)13-s + (0.852 − 0.522i)14-s + (−0.156 + 0.987i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (−0.987 + 0.156i)19-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (−0.852 + 0.522i)3-s + (0.587 + 0.809i)4-s + (0.649 − 0.760i)5-s + (−0.996 + 0.0784i)6-s + (0.522 − 0.852i)7-s + (0.156 + 0.987i)8-s + (0.453 − 0.891i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (0.951 − 0.309i)13-s + (0.852 − 0.522i)14-s + (−0.156 + 0.987i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (−0.987 + 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.800 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647609593 + 0.5480147932i\)
\(L(\frac12)\) \(\approx\) \(1.647609593 + 0.5480147932i\)
\(L(1)\) \(\approx\) \(1.486203914 + 0.4005168575i\)
\(L(1)\) \(\approx\) \(1.486203914 + 0.4005168575i\)

Euler product

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

−27.49593632826498847792250094655, −25.77983319757002108593866052940, −24.91475439559178448476656249193, −24.04100587791023798002567352041, −23.10559866008297384218561411701, −22.29100834130269112117420272545, −21.55640341612149812585529676116, −20.77498371151741002905611681094, −19.048839673044637272079224229362, −18.58747941849042354474951691670, −17.59753736720454121157270413864, −16.26522407065373934197889157941, −15.08002534941258706625118086124, −14.182661276806021809659598333814, −13.11238663774748164785895422787, −12.30032367426199564315852662612, −10.99399862446484075794831538939, −10.83010275049863722958659486947, −9.14193051575550402047254334703, −7.30607288700939194582355623515, −6.14329319771326813349469252472, −5.66966829361087295839088234021, −4.29085987233651917360170454595, −2.548415089425077537838511067647, −1.64851016464817181029011097320, 1.512651388353258244540309771865, 3.67692466952549352373717978718, 4.60732391870350088326554104278, 5.54605863888929740177551281377, 6.423858554222344969391976490639, 7.79477062906871375390849934900, 9.15134350204352620227177459986, 10.6241954393500101096998911141, 11.360871524863649680833521606827, 12.69328608980415241999310457061, 13.3653005266654314623198056769, 14.54756846617328059138133332081, 15.64006918108716975542598867189, 16.654812322176839489171733204097, 17.14691807188302711928117174024, 18.068481723402462267254659936978, 20.15429233218848133973825015612, 20.93447513384455094042078257862, 21.491310850198020752231242275107, 22.612134390465528450869252139195, 23.56963300124573724418119143826, 24.01150380684754863863682175186, 25.27119492185103407168750510434, 26.121355733144179981382821951788, 27.36167487935268270519044805164

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