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Properties

Label 1-191-191.155-r1-0-0
Degree $1$
Conductor $191$
Sign $0.0423 - 0.999i$
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  + (0.789 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.0423 - 0.999i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.0423 - 0.999i)\)

Particular Values

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

Euler product

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

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

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