Properties

Degree 1
Conductor 191
Sign $0.896 + 0.443i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(191\)
\( \varepsilon \)  =  $0.896 + 0.443i$
motivic weight  =  \(0\)
character  :  $\chi_{191} (41, \cdot )$
Sato-Tate  :  $\mu(38)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 191,\ (1:\ ),\ 0.896 + 0.443i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.053576578 + 0.2466588606i$
$L(\frac12,\chi)$  $\approx$  $1.053576578 + 0.2466588606i$
$L(\chi,1)$  $\approx$  0.7981438535 + 0.2037871429i
$L(1,\chi)$  $\approx$  0.7981438535 + 0.2037871429i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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