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Properties

Label 1-191-191.27-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.789 + 0.614i$
Analytic cond. $0.887000$
Root an. cond. $0.887000$
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  + (−0.846 − 0.533i)2-s + (−0.973 + 0.229i)3-s + (0.431 + 0.901i)4-s + (−0.879 − 0.475i)5-s + (0.945 + 0.324i)6-s + (0.309 + 0.951i)7-s + (0.115 − 0.993i)8-s + (0.894 − 0.446i)9-s + (0.490 + 0.871i)10-s + (−0.401 − 0.915i)11-s + (−0.627 − 0.778i)12-s + (0.652 + 0.757i)13-s + (0.245 − 0.969i)14-s + (0.965 + 0.261i)15-s + (−0.627 + 0.778i)16-s + (−0.999 + 0.0330i)17-s + ⋯
L(s)  = 1  + (−0.846 − 0.533i)2-s + (−0.973 + 0.229i)3-s + (0.431 + 0.901i)4-s + (−0.879 − 0.475i)5-s + (0.945 + 0.324i)6-s + (0.309 + 0.951i)7-s + (0.115 − 0.993i)8-s + (0.894 − 0.446i)9-s + (0.490 + 0.871i)10-s + (−0.401 − 0.915i)11-s + (−0.627 − 0.778i)12-s + (0.652 + 0.757i)13-s + (0.245 − 0.969i)14-s + (0.965 + 0.261i)15-s + (−0.627 + 0.778i)16-s + (−0.999 + 0.0330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.789 + 0.614i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.789 + 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02678980070 + 0.07802599042i\)
\(L(\frac12)\) \(\approx\) \(0.02678980070 + 0.07802599042i\)
\(L(1)\) \(\approx\) \(0.3691032701 + 0.02496501159i\)
\(L(1)\) \(\approx\) \(0.3691032701 + 0.02496501159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.846 - 0.533i)T \)
3 \( 1 + (-0.973 + 0.229i)T \)
5 \( 1 + (-0.879 - 0.475i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.401 - 0.915i)T \)
13 \( 1 + (0.652 + 0.757i)T \)
17 \( 1 + (-0.999 + 0.0330i)T \)
19 \( 1 + (0.490 - 0.871i)T \)
23 \( 1 + (-0.909 + 0.416i)T \)
29 \( 1 + (-0.934 + 0.355i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (0.245 + 0.969i)T \)
43 \( 1 + (-0.574 - 0.818i)T \)
47 \( 1 + (-0.213 + 0.976i)T \)
53 \( 1 + (-0.995 + 0.0990i)T \)
59 \( 1 + (-0.461 + 0.887i)T \)
61 \( 1 + (-0.340 + 0.940i)T \)
67 \( 1 + (-0.277 - 0.960i)T \)
71 \( 1 + (0.371 - 0.928i)T \)
73 \( 1 + (0.863 - 0.504i)T \)
79 \( 1 + (-0.934 - 0.355i)T \)
83 \( 1 + (-0.909 - 0.416i)T \)
89 \( 1 + (-0.956 - 0.293i)T \)
97 \( 1 + (0.701 - 0.712i)T \)
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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

−26.78234829465190498715694220306, −26.03463092207428195125535277337, −24.672770934862621171598621718, −23.83260634656143761250823426873, −23.076599221216773708615376245898, −22.51451703066393888452494082232, −20.555140072894167922152951526217, −19.926159519234297704767008919270, −18.57646235389221208155195931347, −18.02976313439078894035406457838, −17.14754272413324930234401426523, −16.0689200515627907414692054306, −15.43987378440582461754405597379, −14.2057240348737142938811413394, −12.754403492586641581592712797384, −11.441806868706182258830640344280, −10.72932385816244091713907738186, −9.987649134061139173430159544, −8.136435812110174097033458383716, −7.41335566158273155971137902285, −6.60740129110192600650687193414, −5.288769660102686754276329901214, −4.00807581777986185462782438107, −1.75000421297557063389714670719, −0.097311193144032378478156268529, 1.62273128569199333117920896404, 3.42879415076715646526732185097, 4.669763768814534689738668403955, 6.037426552816536419474161474441, 7.376064304112699433459267802688, 8.62063961611372332294678290968, 9.31199635677134700603595686619, 11.01143442775005333245827751102, 11.34733837437700995187501979108, 12.21974320459680883639354042105, 13.28455530386452340848614068462, 15.45821362647683444422009395388, 15.97107642702833739973701757920, 16.80048559653941323911299515658, 18.05410733682818629193839509338, 18.59946764160888388448045641967, 19.66564599975225376746581552863, 20.81608006339979398802136659620, 21.67090838636325025296499202279, 22.38974247967886099241355443348, 24.06220107947032520814153926435, 24.21827875437406316176816221853, 25.90831504318519655213590727091, 26.85976756633595830382347363416, 27.62165790986997858383787770836

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