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Properties

Label 1-191-191.34-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.999 + 0.0194i$
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  + (0.746 − 0.665i)2-s + (−0.340 − 0.940i)3-s + (0.115 − 0.993i)4-s + (−0.677 + 0.735i)5-s + (−0.879 − 0.475i)6-s + (0.309 − 0.951i)7-s + (−0.574 − 0.818i)8-s + (−0.768 + 0.639i)9-s + (−0.0165 + 0.999i)10-s + (−0.986 − 0.164i)11-s + (−0.973 + 0.229i)12-s + (−0.277 − 0.960i)13-s + (−0.401 − 0.915i)14-s + (0.922 + 0.386i)15-s + (−0.973 − 0.229i)16-s + (0.0495 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.746 − 0.665i)2-s + (−0.340 − 0.940i)3-s + (0.115 − 0.993i)4-s + (−0.677 + 0.735i)5-s + (−0.879 − 0.475i)6-s + (0.309 − 0.951i)7-s + (−0.574 − 0.818i)8-s + (−0.768 + 0.639i)9-s + (−0.0165 + 0.999i)10-s + (−0.986 − 0.164i)11-s + (−0.973 + 0.229i)12-s + (−0.277 − 0.960i)13-s + (−0.401 − 0.915i)14-s + (0.922 + 0.386i)15-s + (−0.973 − 0.229i)16-s + (0.0495 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.999 + 0.0194i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.999 + 0.0194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01017511783 - 1.048213464i\)
\(L(\frac12)\) \(\approx\) \(0.01017511783 - 1.048213464i\)
\(L(1)\) \(\approx\) \(0.6985955028 - 0.8236286967i\)
\(L(1)\) \(\approx\) \(0.6985955028 - 0.8236286967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.746 - 0.665i)T \)
3 \( 1 + (-0.340 - 0.940i)T \)
5 \( 1 + (-0.677 + 0.735i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.986 - 0.164i)T \)
13 \( 1 + (-0.277 - 0.960i)T \)
17 \( 1 + (0.0495 + 0.998i)T \)
19 \( 1 + (-0.0165 - 0.999i)T \)
23 \( 1 + (0.601 + 0.799i)T \)
29 \( 1 + (-0.518 - 0.854i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (0.245 + 0.969i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.991 - 0.131i)T \)
47 \( 1 + (0.894 - 0.446i)T \)
53 \( 1 + (-0.148 - 0.988i)T \)
59 \( 1 + (0.997 - 0.0660i)T \)
61 \( 1 + (0.965 - 0.261i)T \)
67 \( 1 + (-0.934 - 0.355i)T \)
71 \( 1 + (-0.213 - 0.976i)T \)
73 \( 1 + (0.701 - 0.712i)T \)
79 \( 1 + (-0.518 + 0.854i)T \)
83 \( 1 + (0.601 - 0.799i)T \)
89 \( 1 + (0.431 - 0.901i)T \)
97 \( 1 + (0.371 - 0.928i)T \)
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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

−27.31472049896891627778921279943, −26.71466070149238270169016595803, −25.52997939159067564080959726402, −24.58637744503564990379395452676, −23.64992754014254282708262618516, −22.900063611668590594603763837924, −21.908109835232042516869749505805, −20.93954334567390258943825801368, −20.55771126256073729633198276972, −18.783170789886093127735422592913, −17.57247546305243128124213066581, −16.30389929190089737076603339206, −16.06412197484310600702273697700, −15.03733424600806066465071963393, −14.19434744290083014459480918721, −12.5638339679966585619296162953, −12.01117828097008516260279123411, −10.95980450626352850111460385023, −9.22232336916486595183316272036, −8.494242297562784608364995615808, −7.21939823759395120538136999105, −5.60959005312470006034303864139, −4.99388272904942951483464993335, −4.06330364719428024668906060538, −2.671790601987578775236224002686, 0.65680793989180871467033377714, 2.34370011842524838471107359850, 3.43790911024273966677323413465, 4.84466088560998169041235807900, 6.05930820769045208689118065277, 7.23600021684742708930272821336, 8.03035232512949776100028084110, 10.20040807863225391086835551131, 10.974510076996817218331954281309, 11.65721560949311446862229000706, 13.02719634527887493684920862918, 13.43333846659177660721029011643, 14.73416046161998474030569577589, 15.53316889848159426304817870186, 17.1785793256084248576755009207, 18.16008168711824614241254523124, 19.15967953459389726827683844894, 19.772410391353188924868758425703, 20.79388647894689274405706789509, 22.13689906024036439724375753889, 22.88906605984775125513122525142, 23.76924399315928639203497331498, 24.044983516037391835203947278873, 25.56544001804109642235452139032, 26.72686291694680137809592638732

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