Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $-0.959 - 0.282i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.136 − 0.990i)2-s + (0.519 − 0.854i)3-s + (−0.962 + 0.269i)4-s + (−0.917 − 0.398i)6-s + (0.997 − 0.0682i)7-s + (0.398 + 0.917i)8-s + (−0.460 − 0.887i)9-s + (−0.682 − 0.730i)11-s + (−0.269 + 0.962i)12-s + (−0.816 − 0.576i)13-s + (−0.203 − 0.979i)14-s + (0.854 − 0.519i)16-s + (0.730 + 0.682i)17-s + (−0.816 + 0.576i)18-s + (−0.334 − 0.942i)19-s + ⋯
L(s,χ)  = 1  + (−0.136 − 0.990i)2-s + (0.519 − 0.854i)3-s + (−0.962 + 0.269i)4-s + (−0.917 − 0.398i)6-s + (0.997 − 0.0682i)7-s + (0.398 + 0.917i)8-s + (−0.460 − 0.887i)9-s + (−0.682 − 0.730i)11-s + (−0.269 + 0.962i)12-s + (−0.816 − 0.576i)13-s + (−0.203 − 0.979i)14-s + (0.854 − 0.519i)16-s + (0.730 + 0.682i)17-s + (−0.816 + 0.576i)18-s + (−0.334 − 0.942i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.959 - 0.282i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.959 - 0.282i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $-0.959 - 0.282i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (67, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 235,\ (0:\ ),\ -0.959 - 0.282i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1629127033 - 1.129322847i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1629127033 - 1.129322847i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6680480708 - 0.8179073563i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6680480708 - 0.8179073563i\)

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

−26.66368851804494387902031603393, −25.704506856896630638242586500735, −25.08869314773685326409907524544, −24.00897238847465046745083234719, −23.157521457195633156648306196, −22.067178250049220534392647178723, −21.205028895200156690668866437083, −20.32722384894592298900697516578, −19.06102504490428038476949175037, −18.11597606228628229871647603236, −17.05367183350930209537875368451, −16.29704123056445166336358348191, −15.20616431501545846964993798050, −14.600699797540606769215154425350, −13.89947879413686420520830083461, −12.51207316129941051004918328016, −11.01843347507255801537306370030, −9.863206756292895852427212782589, −9.178629497380494066773826602918, −7.89620840921793951444159209431, −7.40766775169092017295291184964, −5.508546830518741723551046739647, −4.87969469039340679065392940307, −3.79800122652061421858499697006, −2.02634380690736297809175637556, 0.84374968021127352740938208038, 2.19000608816976722428267635412, 3.05655316942580069096039361994, 4.53450166208213718621684276530, 5.79074346656738052963007843655, 7.62345168268868935389822873988, 8.16888061330754973165312254816, 9.21194994802174836589560809898, 10.56022803815975924622061880662, 11.40093932032142263640891935804, 12.52586206127794722447603341199, 13.17278692757051313064754125449, 14.27392114591432503985477421159, 14.94576530044798010861097911808, 16.87988285566924035181399581437, 17.7578576582666289550882772717, 18.5056406497904382995863261425, 19.35351619818915901633639513086, 20.19710253738460129559880656392, 21.04987725070211844318745040970, 21.83696670447782465169008196240, 23.19578501174224160791118276108, 23.94985693682358717436640168670, 24.805827250910655244756565385448, 26.121657775175902343083125425914

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