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Properties

Label 1-4011-4011.2231-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.995 - 0.0951i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.693 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.995 - 0.0951i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (2231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.995 - 0.0951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1236739336 - 2.593189935i\)
\(L(\frac12)\) \(\approx\) \(0.1236739336 - 2.593189935i\)
\(L(1)\) \(\approx\) \(1.210159017 - 1.117984618i\)
\(L(1)\) \(\approx\) \(1.210159017 - 1.117984618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.693 - 0.720i)T \)
5 \( 1 + (0.716 - 0.697i)T \)
11 \( 1 + (0.998 - 0.0550i)T \)
13 \( 1 + (0.909 - 0.416i)T \)
17 \( 1 + (-0.857 - 0.514i)T \)
19 \( 1 + (-0.00551 + 0.999i)T \)
23 \( 1 + (-0.952 + 0.303i)T \)
29 \( 1 + (-0.180 + 0.983i)T \)
31 \( 1 + (-0.904 - 0.426i)T \)
37 \( 1 + (0.904 - 0.426i)T \)
41 \( 1 + (0.789 - 0.614i)T \)
43 \( 1 + (-0.461 - 0.887i)T \)
47 \( 1 + (0.988 + 0.153i)T \)
53 \( 1 + (-0.840 - 0.542i)T \)
59 \( 1 + (-0.480 - 0.876i)T \)
61 \( 1 + (-0.996 - 0.0880i)T \)
67 \( 1 + (-0.992 + 0.120i)T \)
71 \( 1 + (-0.828 - 0.560i)T \)
73 \( 1 + (0.709 - 0.705i)T \)
79 \( 1 + (0.761 - 0.648i)T \)
83 \( 1 + (-0.213 - 0.976i)T \)
89 \( 1 + (-0.782 + 0.622i)T \)
97 \( 1 + (-0.922 - 0.386i)T \)
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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

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

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