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Properties

Label 1-191-191.29-r1-0-0
Degree $1$
Conductor $191$
Sign $-0.684 + 0.728i$
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  + (−0.627 − 0.778i)2-s + (0.601 + 0.799i)3-s + (−0.213 + 0.976i)4-s + (−0.401 − 0.915i)5-s + (0.245 − 0.969i)6-s + (−0.309 − 0.951i)7-s + (0.894 − 0.446i)8-s + (−0.277 + 0.960i)9-s + (−0.461 + 0.887i)10-s + (0.0825 − 0.996i)11-s + (−0.909 + 0.416i)12-s + (−0.956 + 0.293i)13-s + (−0.546 + 0.837i)14-s + (0.490 − 0.871i)15-s + (−0.909 − 0.416i)16-s + (0.991 + 0.131i)17-s + ⋯
L(s)  = 1  + (−0.627 − 0.778i)2-s + (0.601 + 0.799i)3-s + (−0.213 + 0.976i)4-s + (−0.401 − 0.915i)5-s + (0.245 − 0.969i)6-s + (−0.309 − 0.951i)7-s + (0.894 − 0.446i)8-s + (−0.277 + 0.960i)9-s + (−0.461 + 0.887i)10-s + (0.0825 − 0.996i)11-s + (−0.909 + 0.416i)12-s + (−0.956 + 0.293i)13-s + (−0.546 + 0.837i)14-s + (0.490 − 0.871i)15-s + (−0.909 − 0.416i)16-s + (0.991 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01771400135 + 0.04095826930i\)
\(L(\frac12)\) \(\approx\) \(0.01771400135 + 0.04095826930i\)
\(L(1)\) \(\approx\) \(0.6378453820 - 0.1655994344i\)
\(L(1)\) \(\approx\) \(0.6378453820 - 0.1655994344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.627 - 0.778i)T \)
3 \( 1 + (0.601 + 0.799i)T \)
5 \( 1 + (-0.401 - 0.915i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.0825 - 0.996i)T \)
13 \( 1 + (-0.956 + 0.293i)T \)
17 \( 1 + (0.991 + 0.131i)T \)
19 \( 1 + (0.461 + 0.887i)T \)
23 \( 1 + (-0.148 + 0.988i)T \)
29 \( 1 + (-0.115 - 0.993i)T \)
31 \( 1 + (-0.789 + 0.614i)T \)
37 \( 1 + (-0.789 - 0.614i)T \)
41 \( 1 + (-0.546 - 0.837i)T \)
43 \( 1 + (-0.768 + 0.639i)T \)
47 \( 1 + (-0.652 + 0.757i)T \)
53 \( 1 + (-0.922 - 0.386i)T \)
59 \( 1 + (-0.340 - 0.940i)T \)
61 \( 1 + (-0.180 + 0.983i)T \)
67 \( 1 + (0.431 + 0.901i)T \)
71 \( 1 + (-0.0495 + 0.998i)T \)
73 \( 1 + (0.518 - 0.854i)T \)
79 \( 1 + (0.115 - 0.993i)T \)
83 \( 1 + (0.148 + 0.988i)T \)
89 \( 1 + (-0.371 + 0.928i)T \)
97 \( 1 + (-0.999 - 0.0330i)T \)
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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

−26.16868731355164789727027434445, −25.62136961245560125828568114549, −24.824999899690388257966300926072, −23.90345169100917475326080258706, −22.90387437824555546041367585497, −22.08095261032145750507716718471, −20.209737486768913639886363380132, −19.53659748797691997745204075151, −18.44894728537232976614202637253, −18.24564319422165791894468263852, −16.94517569422770881286755427715, −15.44804091628157682109399930784, −14.91248108178369103453636386552, −14.17063533291510511556890656113, −12.69570653806161456569942518943, −11.72520775788424968795689258035, −10.13542413058510210623236703208, −9.26768400012044532266104206879, −8.0755011681136804518693657948, −7.20615349639768312983411784383, −6.50993775286980723612061924558, −5.103091418105909929353171931439, −3.07945584190028968987582517563, −1.95471583718780334282094826777, −0.01758354789679187986759742489, 1.493760286377517765171690609118, 3.31902254489546577517240793549, 3.9274551931000627811636699744, 5.21829497402191453236369372621, 7.52665903369359241930880069611, 8.21218637251626855628239572632, 9.39636497518877616138354192717, 10.01223636879391759741131005878, 11.17019811694922829834681470121, 12.23337859843386672378323833116, 13.414731712760999325442133901177, 14.28144253133826013662965340559, 15.998312421788076730270017688092, 16.55669044041303843601640104551, 17.28043246100890669549872580408, 19.15643952768555859601116204909, 19.5000401423555696033782062631, 20.50613080425505030191190610951, 21.11917412360739605187064718113, 22.07074140703201112763383593225, 23.26070173948801290066047203619, 24.57768394672068094057054864561, 25.59603076602551225158943021592, 26.69008849792442896236079371550, 27.120757543880436841298272796391

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