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Properties

Label 1-191-191.129-r0-0-0
Degree $1$
Conductor $191$
Sign $0.316 + 0.948i$
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  + (−0.213 − 0.976i)2-s + (−0.277 − 0.960i)3-s + (−0.909 + 0.416i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.809 − 0.587i)7-s + (0.601 + 0.799i)8-s + (−0.846 + 0.533i)9-s + (−0.574 + 0.818i)10-s + (−0.986 + 0.164i)11-s + (0.652 + 0.757i)12-s + (0.828 + 0.560i)13-s + (−0.401 + 0.915i)14-s + (−0.518 + 0.854i)15-s + (0.652 − 0.757i)16-s + (0.965 − 0.261i)17-s + ⋯
L(s)  = 1  + (−0.213 − 0.976i)2-s + (−0.277 − 0.960i)3-s + (−0.909 + 0.416i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.809 − 0.587i)7-s + (0.601 + 0.799i)8-s + (−0.846 + 0.533i)9-s + (−0.574 + 0.818i)10-s + (−0.986 + 0.164i)11-s + (0.652 + 0.757i)12-s + (0.828 + 0.560i)13-s + (−0.401 + 0.915i)14-s + (−0.518 + 0.854i)15-s + (0.652 − 0.757i)16-s + (0.965 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1257733392 - 0.09066882953i\)
\(L(\frac12)\) \(\approx\) \(-0.1257733392 - 0.09066882953i\)
\(L(1)\) \(\approx\) \(0.2898754468 - 0.3924266366i\)
\(L(1)\) \(\approx\) \(0.2898754468 - 0.3924266366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.213 - 0.976i)T \)
3 \( 1 + (-0.277 - 0.960i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (0.828 + 0.560i)T \)
17 \( 1 + (0.965 - 0.261i)T \)
19 \( 1 + (-0.574 - 0.818i)T \)
23 \( 1 + (-0.956 + 0.293i)T \)
29 \( 1 + (-0.973 - 0.229i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (0.180 + 0.983i)T \)
47 \( 1 + (-0.148 + 0.988i)T \)
53 \( 1 + (0.701 - 0.712i)T \)
59 \( 1 + (-0.768 - 0.639i)T \)
61 \( 1 + (-0.934 + 0.355i)T \)
67 \( 1 + (-0.627 - 0.778i)T \)
71 \( 1 + (-0.995 + 0.0990i)T \)
73 \( 1 + (-0.461 + 0.887i)T \)
79 \( 1 + (-0.973 + 0.229i)T \)
83 \( 1 + (-0.956 - 0.293i)T \)
89 \( 1 + (-0.724 + 0.689i)T \)
97 \( 1 + (0.997 - 0.0660i)T \)
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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

−27.82759515511938131217880902816, −26.70429550076569849245497126728, −25.95868256788256269931790888025, −25.46102513470582436762700665413, −23.80658825843060412433723011362, −23.052176274025826691367509358421, −22.4601447380220410277166243288, −21.4829784907292442125476993607, −20.1512769805420240974673445473, −18.70974662120843477405648285565, −18.401745682645295522184348374546, −16.821506750038153581689834656542, −16.086180727501010553440859324403, −15.33188981184678117738973216427, −14.74809142295170222061331011837, −13.346156415993702301407774321396, −12.01681164601326670959253373214, −10.52161233163198320616051504895, −10.02528058045707141328713736152, −8.61617399049446446728375161460, −7.7687236315022820399002174951, −6.187010280180149550572022252602, −5.64453925926146375887604626509, −4.077345781109178911699840227, −3.13630818707680271583022849676, 0.139216620943978224147274996707, 1.4932313803521657788482289493, 3.026701723901478555224962016299, 4.27044644043078166901735307907, 5.634627046659182636279391509110, 7.27506983236933612348453372478, 8.135774889051428480786275728783, 9.24262507440604739198351770035, 10.591038491103773717929578563065, 11.53870968918037454475945712200, 12.55216715289520553468350840894, 13.11401658938558605952683932527, 14.00154610371481430186635654617, 15.974814529284940907968156540473, 16.7807768261516199290659328495, 17.87717185249065370700991811968, 18.86714475015374208478523547786, 19.52496671450553790479400797853, 20.35438677849692031796279147345, 21.31025391498518054858684389246, 22.80926300486704532711259928522, 23.34831317630130482692634646193, 24.03953561954152656581775047556, 25.66322836621907197883559658205, 26.25651947534881814173308481495

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