Properties

Degree 1
Conductor 191
Sign $-0.943 - 0.330i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(191\)
\( \varepsilon \)  =  $-0.943 - 0.330i$
motivic weight  =  \(0\)
character  :  $\chi_{191} (9, \cdot )$
Sato-Tate  :  $\mu(95)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 191,\ (0:\ ),\ -0.943 - 0.330i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1456703475 - 0.8573514326i$
$L(\frac12,\chi)$  $\approx$  $0.1456703475 - 0.8573514326i$
$L(\chi,1)$  $\approx$  0.5591342394 - 0.6486539516i
$L(1,\chi)$  $\approx$  0.5591342394 - 0.6486539516i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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