Properties

Degree $1$
Conductor $191$
Sign $0.0423 + 0.999i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.789 − 0.614i)2-s + (−0.677 − 0.735i)3-s + (0.245 − 0.969i)4-s + (0.245 − 0.969i)5-s + (−0.986 − 0.164i)6-s − 7-s + (−0.401 − 0.915i)8-s + (−0.0825 + 0.996i)9-s + (−0.401 − 0.915i)10-s + (−0.546 + 0.837i)11-s + (−0.879 + 0.475i)12-s + (−0.677 + 0.735i)13-s + (−0.789 + 0.614i)14-s + (−0.879 + 0.475i)15-s + (−0.879 − 0.475i)16-s + (0.945 − 0.324i)17-s + ⋯
L(s,χ)  = 1  + (0.789 − 0.614i)2-s + (−0.677 − 0.735i)3-s + (0.245 − 0.969i)4-s + (0.245 − 0.969i)5-s + (−0.986 − 0.164i)6-s − 7-s + (−0.401 − 0.915i)8-s + (−0.0825 + 0.996i)9-s + (−0.401 − 0.915i)10-s + (−0.546 + 0.837i)11-s + (−0.879 + 0.475i)12-s + (−0.677 + 0.735i)13-s + (−0.789 + 0.614i)14-s + (−0.879 + 0.475i)15-s + (−0.879 − 0.475i)16-s + (0.945 − 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0423 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0423 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.0423 + 0.999i$
Motivic weight: \(0\)
Character: $\chi_{191} (122, \cdot )$
Sato-Tate group: $\mu(38)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.0423 + 0.999i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.3602959471 - 0.3453463605i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.3602959471 - 0.3453463605i\)
\(L(\chi,1)\) \(\approx\) \(0.6439158161 - 0.6897036730i\)
\(L(1,\chi)\) \(\approx\) \(0.6439158161 - 0.6897036730i\)

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

−27.12070833428539197395584207905, −26.58177244941618701452496807882, −25.717850022326469487447879205916, −24.74334065750795322660528655611, −23.21311231363094034469326994748, −22.99611026683513768067071519207, −21.862864283733788706620376815267, −21.54395797999500043577999548672, −20.18464354251134376702985385716, −18.687234174630070507105109403158, −17.68707653075409756473738300404, −16.51906482779630712050500046318, −16.00164095381512155972695684689, −14.89828403604166934801894040936, −14.14005793499636967015537302064, −12.801085487815759611633518959825, −11.93145806870336719281145041049, −10.58762507532484757906858374833, −9.93329201994409496820280212354, −8.22017575165514351913029470574, −6.84114766582199916057017998520, −6.00347171953461875995719960936, −5.200545651793102868610115670575, −3.53986461679994418125141955797, −2.98959356823350577670461360694, 0.14017065064913774624382042203, 1.49627431470880249754562334602, 2.73224533076282370103663942779, 4.5256107379989992565977182566, 5.34905842456986570414491289788, 6.41933796458322558316954840014, 7.53409696026513180952237439508, 9.45520216560799770132805614801, 10.13804147165938801965743522682, 11.79196012513012832569751579067, 12.21498736182017140864276014449, 13.22097881957546581979236245096, 13.75819308471140911395430026716, 15.444758610509173914901498153203, 16.38831893939046151963226863816, 17.40707476620888112939051219574, 18.65838450460689722593131516891, 19.52805701944890538306777381520, 20.33826637994397868699129733396, 21.53061773522478634153058925058, 22.34878547453236622835361144170, 23.370294540375748084712952705083, 23.89979577601325974071377354272, 24.91043830324824301858827271151, 25.770902155413496045449294373833

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