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Properties

Label 1-191-191.150-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.605 + 0.795i$
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  + (0.245 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3594471813 + 0.7252328634i\)
\(L(\frac12)\) \(\approx\) \(0.3594471813 + 0.7252328634i\)
\(L(1)\) \(\approx\) \(0.7491136661 + 0.4413985252i\)
\(L(1)\) \(\approx\) \(0.7491136661 + 0.4413985252i\)

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

−27.20302599428712491253643115542, −26.3012644473029842284914277233, −24.571773251951016925179053094000, −23.60652676374253164559073991122, −22.824102224956149374097776997846, −21.75203473526414198981265355159, −20.92762896109862110174055140204, −20.333710759637876453797174361242, −19.38941487620459641331799795381, −18.19324515871414222562996194232, −17.09304826451124363793861813262, −15.91271798166113209042391611869, −14.96067899368090904663269104901, −14.06702013611939312644673047402, −12.721417191653745128962392231384, −11.58792942019711327439232906957, −11.01928169617656442649921681891, −10.03307139619870182602307824898, −8.65339485932972117823413879399, −8.083032464535356368485437327510, −5.576311015977532162506787009, −4.824441440185660055334550241233, −3.836764534331209051661010704118, −2.71894557548831288068152562455, −0.60849680290365003118696421639, 1.83718856824897193300712732978, 3.68114105483690697461452504275, 4.896496562412011992196417496850, 6.17548917002654241706400318429, 7.37667969064368522212866532076, 7.75819879734230575827323106749, 8.84553420383343946446962704028, 10.676858762432870613831831460840, 12.0422975912068582915826351797, 12.5332777509423376643426992425, 14.165456612628460493593595735401, 14.48909398542713761473553041579, 15.64014443667636351231350768243, 16.86699315844520485415210143288, 17.7668474738840013837072371546, 18.5642296950622228199454563434, 19.38544590562225096583699772298, 20.80952922118295273710894317772, 22.03142540307586793219669453736, 23.272588565454344181836301120607, 23.59840308000167272615435229203, 24.312329563184130699419731735293, 25.56089129654033072099167924070, 26.09638871557815427154421289237, 27.42918218765472349020095338717

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