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Properties

Label 1-191-191.26-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.832 + 0.554i$
Analytic cond. $0.887000$
Root an. cond. $0.887000$
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  + (0.701 + 0.712i)2-s + (0.0495 + 0.998i)3-s + (−0.0165 + 0.999i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (−0.809 + 0.587i)7-s + (−0.724 + 0.689i)8-s + (−0.995 + 0.0990i)9-s + (0.431 + 0.901i)10-s + (0.245 − 0.969i)11-s + (−0.999 + 0.0330i)12-s + (0.965 − 0.261i)13-s + (−0.986 − 0.164i)14-s + (−0.277 + 0.960i)15-s + (−0.999 − 0.0330i)16-s + (−0.973 + 0.229i)17-s + ⋯
L(s)  = 1  + (0.701 + 0.712i)2-s + (0.0495 + 0.998i)3-s + (−0.0165 + 0.999i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (−0.809 + 0.587i)7-s + (−0.724 + 0.689i)8-s + (−0.995 + 0.0990i)9-s + (0.431 + 0.901i)10-s + (0.245 − 0.969i)11-s + (−0.999 + 0.0330i)12-s + (0.965 − 0.261i)13-s + (−0.986 − 0.164i)14-s + (−0.277 + 0.960i)15-s + (−0.999 − 0.0330i)16-s + (−0.973 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.832 + 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4975803590 + 1.643870761i\)
\(L(\frac12)\) \(\approx\) \(0.4975803590 + 1.643870761i\)
\(L(1)\) \(\approx\) \(1.005810389 + 1.157670678i\)
\(L(1)\) \(\approx\) \(1.005810389 + 1.157670678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.701 + 0.712i)T \)
3 \( 1 + (0.0495 + 0.998i)T \)
5 \( 1 + (0.945 + 0.324i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.245 - 0.969i)T \)
13 \( 1 + (0.965 - 0.261i)T \)
17 \( 1 + (-0.973 + 0.229i)T \)
19 \( 1 + (0.431 - 0.901i)T \)
23 \( 1 + (0.991 + 0.131i)T \)
29 \( 1 + (0.828 + 0.560i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (-0.909 - 0.416i)T \)
47 \( 1 + (0.997 - 0.0660i)T \)
53 \( 1 + (-0.768 + 0.639i)T \)
59 \( 1 + (-0.213 + 0.976i)T \)
61 \( 1 + (0.652 + 0.757i)T \)
67 \( 1 + (0.922 - 0.386i)T \)
71 \( 1 + (-0.461 - 0.887i)T \)
73 \( 1 + (-0.846 + 0.533i)T \)
79 \( 1 + (0.828 - 0.560i)T \)
83 \( 1 + (0.991 - 0.131i)T \)
89 \( 1 + (0.490 - 0.871i)T \)
97 \( 1 + (0.746 + 0.665i)T \)
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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

−26.60974623216656291135313192313, −25.24877896682454580057311349869, −24.94668719668933740577895700845, −23.539080466113105320294553466539, −23.00015575587755758388841467846, −22.06792725326280475314146680686, −20.703245545855936755996781935915, −20.19609441956481319880051127381, −19.151790219463028067328351161213, −18.18210036286428492555700913652, −17.28545094973775844842568976151, −15.94155805494540948195591238273, −14.41449828211520303594771747527, −13.63867652661278116100306748191, −12.97970218024520865128651744749, −12.21469650323864320979667520391, −10.91049636131368231728769246778, −9.80663365089795961235922292458, −8.8220968865536426859120746241, −6.88718511811968308120037234947, −6.30673682411980460986695186725, −5.025453543545124419156643963859, −3.52935618997475372058903961784, −2.18942839244257242444441599489, −1.18555353400103788261573001747, 2.75740493550965738762407821618, 3.48542901694844412076850709864, 5.00826590326319183526651119376, 5.940134127714949791557958203442, 6.66154311698872649861298930015, 8.67665039782890401120190618039, 9.13354596468956215198051325582, 10.60478600110156341675509368194, 11.60681371539777313492083971979, 13.26304072382753174938040398199, 13.689872681204341429586664263806, 15.00509928197600517125775628240, 15.66450847737085678506263389197, 16.57819100976855217650638702541, 17.46225001862837088435156961726, 18.63416769032280231293225326270, 20.17268696131491765293158477969, 21.22545464001028660457711594448, 22.06535445744197225545545467503, 22.27940534329769079286871112985, 23.5712554573856442208294999231, 24.87532447312996373321519082442, 25.58587766651321937725072301268, 26.26009870722549530967668856785, 27.11578518632515042823351614270

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