Properties

Degree 1
Conductor $ 3 \cdot 7 \cdot 191 $
Sign $-0.995 + 0.0951i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.693 + 0.720i)2-s + (−0.0385 + 0.999i)4-s + (0.716 + 0.697i)5-s + (−0.746 + 0.665i)8-s + (−0.00551 + 0.999i)10-s + (0.998 + 0.0550i)11-s + (0.909 + 0.416i)13-s + (−0.997 − 0.0770i)16-s + (−0.857 + 0.514i)17-s + (−0.00551 − 0.999i)19-s + (−0.724 + 0.689i)20-s + (0.652 + 0.757i)22-s + (−0.952 − 0.303i)23-s + (0.0275 + 0.999i)25-s + (0.329 + 0.944i)26-s + ⋯
L(s,χ)  = 1  + (0.693 + 0.720i)2-s + (−0.0385 + 0.999i)4-s + (0.716 + 0.697i)5-s + (−0.746 + 0.665i)8-s + (−0.00551 + 0.999i)10-s + (0.998 + 0.0550i)11-s + (0.909 + 0.416i)13-s + (−0.997 − 0.0770i)16-s + (−0.857 + 0.514i)17-s + (−0.00551 − 0.999i)19-s + (−0.724 + 0.689i)20-s + (0.652 + 0.757i)22-s + (−0.952 − 0.303i)23-s + (0.0275 + 0.999i)25-s + (0.329 + 0.944i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.995 + 0.0951i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.995 + 0.0951i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4011\)    =    \(3 \cdot 7 \cdot 191\)
\( \varepsilon \)  =  $-0.995 + 0.0951i$
motivic weight  =  \(0\)
character  :  $\chi_{4011} (836, \cdot )$
Sato-Tate  :  $\mu(570)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4011,\ (0:\ ),\ -0.995 + 0.0951i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1236739336 + 2.593189935i$
$L(\frac12,\chi)$  $\approx$  $0.1236739336 + 2.593189935i$
$L(\chi,1)$  $\approx$  1.210159017 + 1.117984618i
$L(1,\chi)$  $\approx$  1.210159017 + 1.117984618i

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

−18.109180620196420132445943199807, −17.73315295669895898236384452454, −16.62578029989621037324845148587, −16.1603132835608315576406755584, −15.31468939669052892650723092161, −14.47438085686984090816049619586, −13.88467892505559042406291362882, −13.43079540776556073466386459314, −12.56134403958279754603794745057, −12.20957159990779903649595765815, −11.26016022518818554899851322469, −10.72209873447580763373565402793, −9.87478178792400437177255713953, −9.17495238941016848989651750232, −8.78857524030727061960617397141, −7.64569865603791254052044279694, −6.52168108495067102259223817631, −5.94555997843157746597478320707, −5.44174123311791602252252976421, −4.43913064306300979030457926946, −3.90026554954033230550907860993, −3.09291380247948808288129156299, −1.89979406203866098378254559676, −1.6034551960893395566341420363, −0.51812964706210259682994600212, 1.40615812583941116091457731191, 2.326810551119421037717441647811, 3.05683470073938177102607192277, 4.08290396254058740145048994057, 4.40386167161683302096128685116, 5.64055372485640203936374496861, 6.268289782847904468749007628647, 6.57421145818923114390813933460, 7.39684403373958235802717762309, 8.28576606853779606259859018266, 9.104491956336821885315244068646, 9.54589389759086277531335673547, 10.810523289375101965782319382024, 11.245627827102090145351901394774, 12.01381676101928823792737070390, 12.94533489025269401892177880779, 13.539568414903876600822420702043, 14.00114628539400582744187906146, 14.741271667908117606333557129173, 15.24782330099226749303074746566, 16.01616650435194862584489147991, 16.78050439484758483368594457252, 17.342606499015591049826649798117, 18.03926097080806466937674050800, 18.46292405861154386928586076111

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