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Properties

Label 1-191-191.154-r0-0-0
Degree $1$
Conductor $191$
Sign $0.695 + 0.718i$
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  + (−0.986 + 0.164i)2-s + (0.546 + 0.837i)3-s + (0.945 − 0.324i)4-s + (0.945 − 0.324i)5-s + (−0.677 − 0.735i)6-s + 7-s + (−0.879 + 0.475i)8-s + (−0.401 + 0.915i)9-s + (−0.879 + 0.475i)10-s + (0.245 + 0.969i)11-s + (0.789 + 0.614i)12-s + (0.546 − 0.837i)13-s + (−0.986 + 0.164i)14-s + (0.789 + 0.614i)15-s + (0.789 − 0.614i)16-s + (−0.0825 − 0.996i)17-s + ⋯
L(s)  = 1  + (−0.986 + 0.164i)2-s + (0.546 + 0.837i)3-s + (0.945 − 0.324i)4-s + (0.945 − 0.324i)5-s + (−0.677 − 0.735i)6-s + 7-s + (−0.879 + 0.475i)8-s + (−0.401 + 0.915i)9-s + (−0.879 + 0.475i)10-s + (0.245 + 0.969i)11-s + (0.789 + 0.614i)12-s + (0.546 − 0.837i)13-s + (−0.986 + 0.164i)14-s + (0.789 + 0.614i)15-s + (0.789 − 0.614i)16-s + (−0.0825 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057330046 + 0.4483890211i\)
\(L(\frac12)\) \(\approx\) \(1.057330046 + 0.4483890211i\)
\(L(1)\) \(\approx\) \(0.9930386065 + 0.2854131112i\)
\(L(1)\) \(\approx\) \(0.9930386065 + 0.2854131112i\)

Euler product

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

−26.741540056625598568696175875375, −26.01703553353090793717105255534, −25.24680428993031874407422586727, −24.34357844334720877638937602469, −23.688982418548868827927730323769, −21.63157074276994461454854537086, −21.18830523697366906978308208330, −20.0986406599667060677317614963, −18.94208764046957233228183263208, −18.47213246908142770322500422250, −17.45699302606428355297831246268, −16.80703841751498993823925207494, −15.138400013204216272408207316584, −14.19470685234532994970795098775, −13.298304195581136289004971337188, −11.8957625002895578102293572914, −11.03834300383837204927501780156, −9.84292665635230021752671921444, −8.58742006649748360115363399438, −8.14628637758698897013540898538, −6.63975529475808904568279309853, −5.99610267853578667584200959890, −3.613109793604131781962116112482, −2.091956986747541148704509778749, −1.497397342095954953415286047346, 1.63035264157723008687192863488, 2.672158187241565808083954056741, 4.6077765397712404939086130654, 5.6212401404673823825715584027, 7.157811652744349895406863888878, 8.45352405997114689245371525859, 9.03337326202224937930535102820, 10.20370924666922117876852487932, 10.74938875217944636022596433320, 12.21244115617879983390347241546, 13.82739926702164387578466884274, 14.71292457468792244740552665192, 15.61017148926773705634661937625, 16.64545624961085709623105773024, 17.66505778442838992296384744491, 18.16409946373626063175709946275, 19.80874298874306332755085416967, 20.50088657122537864215232809899, 21.069058167967791339775615011531, 22.14673232539306303176215947015, 23.66138521813936834459001896780, 25.08474191155497304019460689835, 25.21114241673054521465481578319, 26.24860119196674510863664961096, 27.33641401458139440988838925666

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