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Properties

Label 1-191-191.102-r0-0-0
Degree $1$
Conductor $191$
Sign $0.702 - 0.711i$
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  + (−0.0165 − 0.999i)2-s + (−0.995 − 0.0990i)3-s + (−0.999 + 0.0330i)4-s + (0.789 − 0.614i)5-s + (−0.0825 + 0.996i)6-s + (0.309 + 0.951i)7-s + (0.0495 + 0.998i)8-s + (0.980 + 0.197i)9-s + (−0.627 − 0.778i)10-s + (−0.879 + 0.475i)11-s + (0.997 + 0.0660i)12-s + (0.863 + 0.504i)13-s + (0.945 − 0.324i)14-s + (−0.846 + 0.533i)15-s + (0.997 − 0.0660i)16-s + (0.894 + 0.446i)17-s + ⋯
L(s)  = 1  + (−0.0165 − 0.999i)2-s + (−0.995 − 0.0990i)3-s + (−0.999 + 0.0330i)4-s + (0.789 − 0.614i)5-s + (−0.0825 + 0.996i)6-s + (0.309 + 0.951i)7-s + (0.0495 + 0.998i)8-s + (0.980 + 0.197i)9-s + (−0.627 − 0.778i)10-s + (−0.879 + 0.475i)11-s + (0.997 + 0.0660i)12-s + (0.863 + 0.504i)13-s + (0.945 − 0.324i)14-s + (−0.846 + 0.533i)15-s + (0.997 − 0.0660i)16-s + (0.894 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8166470820 - 0.3411124381i\)
\(L(\frac12)\) \(\approx\) \(0.8166470820 - 0.3411124381i\)
\(L(1)\) \(\approx\) \(0.7900210815 - 0.3248222732i\)
\(L(1)\) \(\approx\) \(0.7900210815 - 0.3248222732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.0165 - 0.999i)T \)
3 \( 1 + (-0.995 - 0.0990i)T \)
5 \( 1 + (0.789 - 0.614i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (0.863 + 0.504i)T \)
17 \( 1 + (0.894 + 0.446i)T \)
19 \( 1 + (-0.627 + 0.778i)T \)
23 \( 1 + (0.965 - 0.261i)T \)
29 \( 1 + (0.371 - 0.928i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.677 + 0.735i)T \)
41 \( 1 + (0.945 + 0.324i)T \)
43 \( 1 + (0.652 - 0.757i)T \)
47 \( 1 + (0.991 + 0.131i)T \)
53 \( 1 + (0.180 + 0.983i)T \)
59 \( 1 + (-0.909 + 0.416i)T \)
61 \( 1 + (-0.148 - 0.988i)T \)
67 \( 1 + (0.701 + 0.712i)T \)
71 \( 1 + (-0.574 - 0.818i)T \)
73 \( 1 + (0.431 + 0.901i)T \)
79 \( 1 + (0.371 + 0.928i)T \)
83 \( 1 + (0.965 + 0.261i)T \)
89 \( 1 + (-0.518 + 0.854i)T \)
97 \( 1 + (0.115 - 0.993i)T \)
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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

−27.104192342371298604822606478624, −26.20051659542277562805979039406, −25.40717050566441393816444680191, −24.18776788747442606189023233711, −23.29101606010246737695812649089, −22.893303856375287105140293875124, −21.62453481591780179267485727618, −20.98979960487805622096085850061, −19.02932598154856755789257229948, −18.04404060298054369236275208237, −17.5746596901422532773866159470, −16.57010974650845445848988470182, −15.77167429237666951071075333191, −14.54856493763404886910864104580, −13.54470810636431041148747465458, −12.801981418423946752508674717026, −10.81873198693425578201624935996, −10.528865240995187482648477964830, −9.14341393853498276195499594121, −7.59477055599382798554011317804, −6.79413962062107532303814389802, −5.70958709406270651787599605059, −4.96241750299257361740874065338, −3.43898768630084539929201487386, −0.98914029497112577084891387559, 1.30964551376681038899474309953, 2.32819030561445590360308474215, 4.26137096224205637632870350025, 5.34017021153414360417221379541, 6.01218989256509986411255989628, 8.04056233574323749284993492228, 9.17706248772785833708309105274, 10.20025981435704537607740113822, 11.08003721599827971730316354786, 12.30786927288327832202357038621, 12.709519112466367872638792778076, 13.83680551043261093253238074927, 15.29484990468108769016353555212, 16.68019835321316844794354596471, 17.4551789719553755098708274795, 18.48313694412447131638964517739, 18.88970064543414250988406705148, 20.9008019166224978886323651319, 21.00758949394599687500233456209, 21.97658028144962500444340972115, 23.08284824581481784959825203481, 23.79931014198249722353658238594, 25.03898325842688086075917717586, 26.08780876909803771778704246626, 27.59900197612611794786825419873

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