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Properties

Label 1-191-191.66-r1-0-0
Degree $1$
Conductor $191$
Sign $0.206 - 0.978i$
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  + (−0.0825 − 0.996i)2-s + (−0.879 − 0.475i)3-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s − 7-s + (0.245 + 0.969i)8-s + (0.546 + 0.837i)9-s + (0.245 + 0.969i)10-s + (−0.789 − 0.614i)11-s + (0.945 + 0.324i)12-s + (−0.879 + 0.475i)13-s + (0.0825 + 0.996i)14-s + (0.945 + 0.324i)15-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + ⋯
L(s)  = 1  + (−0.0825 − 0.996i)2-s + (−0.879 − 0.475i)3-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s − 7-s + (0.245 + 0.969i)8-s + (0.546 + 0.837i)9-s + (0.245 + 0.969i)10-s + (−0.789 − 0.614i)11-s + (0.945 + 0.324i)12-s + (−0.879 + 0.475i)13-s + (0.0825 + 0.996i)14-s + (0.945 + 0.324i)15-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1787047676 - 0.1449621679i\)
\(L(\frac12)\) \(\approx\) \(0.1787047676 - 0.1449621679i\)
\(L(1)\) \(\approx\) \(0.3304906364 - 0.1959996023i\)
\(L(1)\) \(\approx\) \(0.3304906364 - 0.1959996023i\)

Euler product

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

−26.9065172613590765120565134708, −26.32780371552807208269662319411, −25.147184465240352305775094186409, −23.9468743920617544952915572830, −23.3226574064302349919772503650, −22.55689352381239826985754917117, −21.85633557974314162799564746672, −20.222869105489641771488985571616, −19.212786469382270304710217839483, −18.05636502186613870831913372137, −17.22492946357206994262867302266, −16.10823301503731568326370924533, −15.67201881923684991331441611615, −14.90330747362598535395647904357, −13.095575221486876902778420677588, −12.48013096120115936942098362145, −11.08474620646095745059431135750, −9.888422044964921253895085561063, −9.03803144006619699494364852347, −7.41861148805621048140162472508, −6.869695123540013252937053542295, −5.363115242101714584771328145720, −4.65644613127314724341494133948, −3.362955361247485730936555142627, −0.31707901197731867098365500866, 0.3246273095246434754683069451, 2.1811183397826768670187619626, 3.56534896025440332775100712154, 4.711381028242304622281983473961, 6.09766906640330440994535634604, 7.4137600713123686110936893331, 8.51726939563969424692444384292, 10.04360852307203376650188132601, 10.84981290061438771095050751579, 11.79474090749872728631683275912, 12.64066653996495546195878174122, 13.30244273306534839053745519614, 14.86623140742888948847646654218, 16.29284271467614907885911984829, 16.95850638000192869300754893509, 18.39478503677641160848830253706, 19.01083948715396654681314520798, 19.58754664294099189559207302478, 20.89665616812767331533257294048, 22.21151906267322115515978059249, 22.56474834540251476273101070206, 23.60915505878484444546725131732, 24.30901717643794330890548110469, 26.11717759148888513049767073020, 26.83895748100982643142386125893

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