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Properties

Label 1-191-191.9-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.943 - 0.330i$
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  + (−0.148 − 0.988i)2-s + (−0.627 − 0.778i)3-s + (−0.956 + 0.293i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (0.309 − 0.951i)7-s + (0.431 + 0.901i)8-s + (−0.213 + 0.976i)9-s + (0.180 − 0.983i)10-s + (0.245 − 0.969i)11-s + (0.828 + 0.560i)12-s + (0.0495 − 0.998i)13-s + (−0.986 − 0.164i)14-s + (−0.340 − 0.940i)15-s + (0.828 − 0.560i)16-s + (−0.518 − 0.854i)17-s + ⋯
L(s)  = 1  + (−0.148 − 0.988i)2-s + (−0.627 − 0.778i)3-s + (−0.956 + 0.293i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (0.309 − 0.951i)7-s + (0.431 + 0.901i)8-s + (−0.213 + 0.976i)9-s + (0.180 − 0.983i)10-s + (0.245 − 0.969i)11-s + (0.828 + 0.560i)12-s + (0.0495 − 0.998i)13-s + (−0.986 − 0.164i)14-s + (−0.340 − 0.940i)15-s + (0.828 − 0.560i)16-s + (−0.518 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.943 - 0.330i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.943 - 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1456703475 - 0.8573514326i\)
\(L(\frac12)\) \(\approx\) \(0.1456703475 - 0.8573514326i\)
\(L(1)\) \(\approx\) \(0.5591342394 - 0.6486539516i\)
\(L(1)\) \(\approx\) \(0.5591342394 - 0.6486539516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.148 - 0.988i)T \)
3 \( 1 + (-0.627 - 0.778i)T \)
5 \( 1 + (0.945 + 0.324i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.245 - 0.969i)T \)
13 \( 1 + (0.0495 - 0.998i)T \)
17 \( 1 + (-0.518 - 0.854i)T \)
19 \( 1 + (0.180 + 0.983i)T \)
23 \( 1 + (-0.724 - 0.689i)T \)
29 \( 1 + (-0.277 + 0.960i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (0.115 - 0.993i)T \)
47 \( 1 + (0.371 + 0.928i)T \)
53 \( 1 + (0.997 - 0.0660i)T \)
59 \( 1 + (0.746 - 0.665i)T \)
61 \( 1 + (-0.973 - 0.229i)T \)
67 \( 1 + (0.652 + 0.757i)T \)
71 \( 1 + (0.701 - 0.712i)T \)
73 \( 1 + (-0.768 - 0.639i)T \)
79 \( 1 + (-0.277 - 0.960i)T \)
83 \( 1 + (-0.724 + 0.689i)T \)
89 \( 1 + (0.980 + 0.197i)T \)
97 \( 1 + (0.863 - 0.504i)T \)
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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

−27.598385168748329204525572849147, −26.282726504891881755711365713798, −25.77768886155983512290650982397, −24.64973793206770466537638408594, −23.88316536918084222117948477754, −22.74738026662641321495159992105, −21.70139996176092531401275866934, −21.44135367671495686920904750199, −19.82636267886651121644212340436, −18.27540885820169352127891594728, −17.61803428945090239392897632637, −16.95155146106454959306068462529, −15.82641253377497411208513989230, −15.09285892876952116663511816168, −14.12841943479581520408877016766, −12.85103448900516403639340057134, −11.70580838767377665936324541623, −10.22272694039778629458813304655, −9.31331316188532116461417867773, −8.73686174386716445820710372207, −6.89959620965925289255556084820, −5.94748166199854817424538708095, −5.09780458001861913443145962438, −4.16904324966172959198071243676, −1.84706993876375333119273083432, 0.81944602898365185413935852722, 2.00471590394842086434248420347, 3.3647244700921044090117343128, 5.01352221527557979324705171741, 6.0604897980547182432835555085, 7.43529116994956992324894624002, 8.570030715027894778771741891018, 10.11162373606000054856827347105, 10.734150693156529567262134821725, 11.66359173697260584437835950210, 12.89032252042167091733578666030, 13.66833268304337165556379656535, 14.27896492319825843005915715136, 16.55748154696919314741516664630, 17.249346128366455970054352689618, 18.19243582800368374683780478699, 18.70308643219482053218710164795, 20.06518010598279898382529376469, 20.74767449593895879370645713134, 22.189816612426366139030494335006, 22.45416731304255877218166631433, 23.68489965445824992623318471092, 24.66848538810235120910536014373, 25.7988925163539414615069845781, 26.9669176801849278712253342410

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