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Properties

Label 1-191-191.166-r1-0-0
Degree $1$
Conductor $191$
Sign $-0.877 - 0.478i$
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  + (0.546 + 0.837i)2-s + (0.945 + 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s − 7-s + (−0.986 + 0.164i)8-s + (0.789 + 0.614i)9-s + (−0.986 + 0.164i)10-s + (0.0825 + 0.996i)11-s + (−0.677 + 0.735i)12-s + (0.945 − 0.324i)13-s + (−0.546 − 0.837i)14-s + (−0.677 + 0.735i)15-s + (−0.677 − 0.735i)16-s + (−0.879 − 0.475i)17-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + (0.945 + 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s − 7-s + (−0.986 + 0.164i)8-s + (0.789 + 0.614i)9-s + (−0.986 + 0.164i)10-s + (0.0825 + 0.996i)11-s + (−0.677 + 0.735i)12-s + (0.945 − 0.324i)13-s + (−0.546 − 0.837i)14-s + (−0.677 + 0.735i)15-s + (−0.677 − 0.735i)16-s + (−0.879 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.877 - 0.478i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ -0.877 - 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5189654359 + 2.035476417i\)
\(L(\frac12)\) \(\approx\) \(-0.5189654359 + 2.035476417i\)
\(L(1)\) \(\approx\) \(0.8461217385 + 1.179991086i\)
\(L(1)\) \(\approx\) \(0.8461217385 + 1.179991086i\)

Euler product

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

−26.37809046213005293304047518576, −25.21954262183112493321155939714, −24.090561961456499234268396645529, −23.71112391966499215436470107461, −22.293000161999069457594533423424, −21.37609746388259096026302087562, −20.362081555965223582140554228514, −19.739109575187437527540255077613, −19.067546060876115612570821268269, −18.030198453108512203747105299936, −16.13081397083082685052812834829, −15.61850814877804317933562640596, −14.04514273025240147684658666019, −13.44274434805383396103137905322, −12.63992142946728569939020895618, −11.68390112324542013882846439907, −10.30424846187936611116567570347, −9.01134803603026031843051633646, −8.61777691634129231100015041113, −6.807831623908017307113796301686, −5.54345220519262910970413756556, −3.89563346621752688446552432804, −3.396326226763426109384740321138, −1.81617352610142844314750712827, −0.53007417409992785796640969633, 2.59639065541486572791833750073, 3.515563582825844240645553802989, 4.42343557647910760665310933142, 6.15965714912249549013905138635, 7.12124786352432443752698345701, 7.97922183123923434142234109608, 9.24916777168014436632616401145, 10.19750019183595290655490829262, 11.79215501195772319935819850487, 13.08392795019552625467246629458, 13.82693519182520477831674550364, 14.820177393015861196640497184391, 15.702016090516716782536243311591, 16.08940014369077911566999164481, 17.83045841322251747035234811415, 18.63513536752725237748114467000, 19.88860441238142689168681191575, 20.68799673827840793683368836195, 22.16108611687839064337594552983, 22.49610146551922332615486179718, 23.51178289144348144011257763099, 24.8172229905529684028058456094, 25.617304715537200897901891976368, 26.20210572864509943662794188803, 26.888321000261871486339766986359

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