Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.771 + 0.636i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.987 + 0.156i)2-s + (−0.760 − 0.649i)3-s + (0.951 − 0.309i)4-s + (−0.972 + 0.233i)5-s + (0.852 + 0.522i)6-s + (−0.649 − 0.760i)7-s + (−0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (−0.587 + 0.809i)13-s + (0.760 + 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯
L(s,χ)  = 1  + (−0.987 + 0.156i)2-s + (−0.760 − 0.649i)3-s + (0.951 − 0.309i)4-s + (−0.972 + 0.233i)5-s + (0.852 + 0.522i)6-s + (−0.649 − 0.760i)7-s + (−0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (0.923 − 0.382i)10-s + (−0.923 − 0.382i)12-s + (−0.587 + 0.809i)13-s + (0.760 + 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.771 + 0.636i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.771 + 0.636i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.771 + 0.636i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (46, \cdot )$
Sato-Tate  :  $\mu(80)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.771 + 0.636i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3029205908 + 0.1088186488i$
$L(\frac12,\chi)$  $\approx$  $0.3029205908 + 0.1088186488i$
$L(\chi,1)$  $\approx$  0.4145338689 + 0.009558860636i
$L(1,\chi)$  $\approx$  0.4145338689 + 0.009558860636i

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

−27.16352372810660264007458425284, −26.55332274241053141256911828783, −25.26338684564864832714000338034, −24.391587558742538405066796919800, −23.10336511970858004968928347032, −22.31929826262074039425400776708, −21.16450367102697146091697384678, −20.27004289462650587352013370480, −19.218355566812313633732486297294, −18.448342445551802448650588150470, −17.217552022488677416847866787413, −16.46653311610110056403566301391, −15.55015779029210134334908952129, −15.03987969295289030809032086240, −12.43414228995656194743528573519, −12.20198543874098890720956227497, −10.95206298228806860883316683922, −10.08563744972077964524742198741, −9.036491532180115548030666619810, −8.00886490316367341280108306878, −6.687788263121618651760597837678, −5.56753171947756485999199678377, −4.010948106762431639924553976794, −2.741987283379357875251946288681, −0.49256554098791508795186363538, 1.00027034968787332282970553050, 2.75191677567752906465500858648, 4.51005311831426615169760927123, 6.22854117286433843370321269904, 7.12364306626393166329908656375, 7.67023962143183546923406538285, 9.14338634468944390783297891906, 10.4360350345532717461313276658, 11.280652140671308457298616337484, 12.07250169620815682199253738211, 13.30447214550308105769091069323, 14.81368518529759749231615050834, 16.00866716720419711846347657647, 16.68900139574942893833214073182, 17.540703623338564180711054438094, 18.6670014158529870015219119671, 19.45134190025758789427088733449, 19.91582382504095436221095287978, 21.57633925364928149984041251911, 22.844107083977950431772037149692, 23.692587717899647733063177463981, 24.22664712546890864173018333654, 25.54562677205370378365759829604, 26.45262116652559061011827866419, 27.29459924170667168688235672217

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