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Properties

Label 1-191-191.30-r0-0-0
Degree $1$
Conductor $191$
Sign $0.496 + 0.867i$
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  + (−0.677 − 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (0.546 − 0.837i)6-s + 7-s + (0.789 − 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.789 − 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.986 + 0.164i)12-s + (0.245 − 0.969i)13-s + (−0.677 − 0.735i)14-s + (−0.986 + 0.164i)15-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + ⋯
L(s)  = 1  + (−0.677 − 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (0.546 − 0.837i)6-s + 7-s + (0.789 − 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.789 − 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.986 + 0.164i)12-s + (0.245 − 0.969i)13-s + (−0.677 − 0.735i)14-s + (−0.986 + 0.164i)15-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.496 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8317384261 + 0.4823993510i\)
\(L(\frac12)\) \(\approx\) \(0.8317384261 + 0.4823993510i\)
\(L(1)\) \(\approx\) \(0.8767779404 + 0.2145090176i\)
\(L(1)\) \(\approx\) \(0.8767779404 + 0.2145090176i\)

Euler product

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

−26.90554929527518157212957596022, −25.79456409064675652884650899107, −24.74653981382639138501499101422, −24.38573657478269867512766207251, −23.69798320050963531667140251339, −22.51917725791329114301071418335, −20.682542171888047292263010398368, −20.12423514081560143089119048492, −19.04494702165781691450137281053, −18.14391359348156956731722243303, −17.25959481430761985744563735371, −16.58660859726654428823919829119, −15.22347577443749661985329051876, −14.189799170461498935556416557504, −13.467815581490894385795277528435, −11.96375990619532115720511677159, −11.23420764995833924578183495474, −9.10761417210914011928425447285, −8.96698435573682781658999679500, −7.621596288488014465255281682516, −6.89609114618245007430587229680, −5.52336981991200344484851377433, −4.43192521655265464081365423718, −1.961411502176468751704984504804, −1.0409459016167637839091486713, 1.76140275137834367853901509262, 3.254488900665726780779997829760, 3.91654707886235529465620899222, 5.54478768152124291604134725149, 7.32802371651350835438120316184, 8.35919569685841319959404643366, 9.31153037148382958951307192564, 10.57628693131257183726574719531, 10.963837453119661033558791581411, 11.956356180112161638174427115064, 13.6409818896072433092274678413, 14.65232725092382729289023545363, 15.49776602031578677019217219502, 16.866481752359192605095765542919, 17.627959606381467847886793992723, 18.64068650449171911112862939879, 19.72938529492818343611638265617, 20.466332004223123111858295109587, 21.524950862895047696030111883985, 22.10292030109430805694783376598, 23.00665653993587041182369543591, 24.781215413932675830431608942914, 25.70547001093066362170387930785, 26.62388942680866174868993763618, 27.36929572179274768527029325077

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