Properties

Degree $1$
Conductor $191$
Sign $0.206 + 0.978i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.206 + 0.978i$
Motivic weight: \(0\)
Character: $\chi_{191} (55, \cdot )$
Sato-Tate group: $\mu(38)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.206 + 0.978i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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