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Properties

Label 1-191-191.5-r0-0-0
Degree $1$
Conductor $191$
Sign $0.576 + 0.817i$
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  + (−0.879 − 0.475i)2-s + (−0.986 − 0.164i)3-s + (0.546 + 0.837i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + 7-s + (−0.0825 − 0.996i)8-s + (0.945 + 0.324i)9-s + (−0.0825 − 0.996i)10-s + (−0.677 + 0.735i)11-s + (−0.401 − 0.915i)12-s + (−0.986 + 0.164i)13-s + (−0.879 − 0.475i)14-s + (−0.401 − 0.915i)15-s + (−0.401 + 0.915i)16-s + (0.245 − 0.969i)17-s + ⋯
L(s)  = 1  + (−0.879 − 0.475i)2-s + (−0.986 − 0.164i)3-s + (0.546 + 0.837i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + 7-s + (−0.0825 − 0.996i)8-s + (0.945 + 0.324i)9-s + (−0.0825 − 0.996i)10-s + (−0.677 + 0.735i)11-s + (−0.401 − 0.915i)12-s + (−0.986 + 0.164i)13-s + (−0.879 − 0.475i)14-s + (−0.401 − 0.915i)15-s + (−0.401 + 0.915i)16-s + (0.245 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.576 + 0.817i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.576 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5142438074 + 0.2665403811i\)
\(L(\frac12)\) \(\approx\) \(0.5142438074 + 0.2665403811i\)
\(L(1)\) \(\approx\) \(0.6036233832 + 0.06090590167i\)
\(L(1)\) \(\approx\) \(0.6036233832 + 0.06090590167i\)

Euler product

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

−27.077304558326907022905302600999, −26.198916933683072902652782572659, −24.834815751038925649004085538761, −24.01870728596691990809763704732, −23.79163365769112531827354292336, −22.044669148248677889127631627345, −21.18768503009100797768286256639, −20.27484593853128667527525156412, −18.9243326360026219213571668071, −17.91349718987192506036573608523, −17.22747851868624084262671975253, −16.62950607394917317503875294150, −15.569260865674758658865349490247, −14.51156069656574418053071808501, −13.054842471476192399928389848248, −11.854023743795908030164475685660, −10.80149162460571611666551025936, −10.03378668309590464867052564622, −8.773128909932473487891297828556, −7.84319830060430155117784094019, −6.47147348405160707841568674450, −5.36663554224121260732021476317, −4.77614147398583000365203784812, −2.07541132134891834879685826008, −0.70899977638790780504221535459, 1.547732895892808403352231795640, 2.59802788012342202178488629694, 4.53230718791923292910738948895, 5.83938664570510266184544149766, 7.245943738572283289049628778496, 7.7417134961995941314280190, 9.70966406280849091625242082696, 10.21732557864005737368933542880, 11.35557565338283549810589020697, 11.9494513749238875592669620706, 13.20940253485927867203638865643, 14.635161147120986546696182705475, 15.81722602309335595572616955115, 17.08136530892918855472995900029, 17.704904776217367777338321178924, 18.32573877881960203075645852269, 19.19783738652170058493453309586, 20.734683454293661118317937718971, 21.372242609286238851719462644803, 22.33663270559369189026305426126, 23.29542708944445978557405209167, 24.615885929179488314654321580880, 25.351985260876841495486815264084, 26.70073028102717763568354912926, 27.20067775572241387226658159742

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