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Properties

Label 1-191-191.14-r1-0-0
Degree $1$
Conductor $191$
Sign $0.896 - 0.443i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
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  + (0.245 − 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 − 0.475i)4-s + (−0.879 − 0.475i)5-s + (0.945 + 0.324i)6-s − 7-s + (−0.677 + 0.735i)8-s + (−0.986 − 0.164i)9-s + (−0.677 + 0.735i)10-s + (0.401 + 0.915i)11-s + (0.546 − 0.837i)12-s + (−0.0825 − 0.996i)13-s + (−0.245 + 0.969i)14-s + (0.546 − 0.837i)15-s + (0.546 + 0.837i)16-s + (0.789 − 0.614i)17-s + ⋯
L(s)  = 1  + (0.245 − 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 − 0.475i)4-s + (−0.879 − 0.475i)5-s + (0.945 + 0.324i)6-s − 7-s + (−0.677 + 0.735i)8-s + (−0.986 − 0.164i)9-s + (−0.677 + 0.735i)10-s + (0.401 + 0.915i)11-s + (0.546 − 0.837i)12-s + (−0.0825 − 0.996i)13-s + (−0.245 + 0.969i)14-s + (0.546 − 0.837i)15-s + (0.546 + 0.837i)16-s + (0.789 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053576578 - 0.2466588606i\)
\(L(\frac12)\) \(\approx\) \(1.053576578 - 0.2466588606i\)
\(L(1)\) \(\approx\) \(0.7981438535 - 0.2037871429i\)
\(L(1)\) \(\approx\) \(0.7981438535 - 0.2037871429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.245 - 0.969i)T \)
3 \( 1 + (-0.0825 + 0.996i)T \)
5 \( 1 + (-0.879 - 0.475i)T \)
7 \( 1 - T \)
11 \( 1 + (0.401 + 0.915i)T \)
13 \( 1 + (-0.0825 - 0.996i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
19 \( 1 + (0.677 + 0.735i)T \)
23 \( 1 + (-0.677 - 0.735i)T \)
29 \( 1 + (-0.546 + 0.837i)T \)
31 \( 1 + (0.986 + 0.164i)T \)
37 \( 1 + (0.986 - 0.164i)T \)
41 \( 1 + (-0.245 - 0.969i)T \)
43 \( 1 + (0.945 + 0.324i)T \)
47 \( 1 + (0.401 + 0.915i)T \)
53 \( 1 + (0.401 + 0.915i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (-0.789 - 0.614i)T \)
67 \( 1 + (0.789 + 0.614i)T \)
71 \( 1 + (-0.245 - 0.969i)T \)
73 \( 1 + (0.401 - 0.915i)T \)
79 \( 1 + (0.546 + 0.837i)T \)
83 \( 1 + (0.677 - 0.735i)T \)
89 \( 1 + (-0.945 + 0.324i)T \)
97 \( 1 + (-0.986 + 0.164i)T \)
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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

−26.394911230199017166890324434998, −26.077112035924558203207574716443, −24.85229186826911612513165341216, −23.98724277710614071042440884024, −23.37253913234476809305578656617, −22.5060821265041533543553124373, −21.669480913465986952646105711209, −19.6889185527549186777921923745, −19.073599519877006474842093899130, −18.36504868800703736803995293968, −17.00288738537065214901008442567, −16.30888841945241420496993630091, −15.24342674719958163140016868468, −14.07592679140422819229377115623, −13.41350619922660836618465935430, −12.19886895082465747565510285337, −11.46097652216819678711374014964, −9.58788439748934695638856189139, −8.37972747225591198901448620769, −7.459372254845338453824144786751, −6.57322574376259963366361846330, −5.8203202182543056659403627143, −3.99345973044371491585685107423, −2.98724749433376744118590422743, −0.619574015937129261106427763062, 0.705774670679773116710578036041, 2.9042237670663487626247237682, 3.75908300454199265491727553681, 4.726438743935224642184183795348, 5.83433687474268562097976759441, 7.80020890400690495834363678258, 9.15342725747564839671131723579, 9.86412052608858958569491570801, 10.77782881724398622620675530774, 12.15150498836832839767590966904, 12.45955963713157167938565446010, 14.04021661510795042671618675382, 15.09413858768862000987940765081, 15.98466256981575238764901569582, 16.97659001187673427151451794573, 18.31788906151524676460495952349, 19.52452670338689269639738855029, 20.279581221817785940242301346650, 20.71619127336091281237735785715, 22.23521451245144182466677741271, 22.69513656715430934435994006948, 23.34955953271461543168955161576, 24.93046009872828809822340085141, 26.1413571185160277187546159909, 27.186235717959681578198110979196

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