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Properties

Label 1-4011-4011.773-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.411 - 0.911i$
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  + (−0.930 − 0.366i)2-s + (0.731 + 0.681i)4-s + (−0.754 + 0.656i)5-s + (−0.431 − 0.901i)8-s + (0.942 − 0.335i)10-s + (0.962 − 0.272i)11-s + (−0.0495 + 0.998i)13-s + (0.0715 + 0.997i)16-s + (−0.480 + 0.876i)17-s + (0.942 + 0.335i)19-s + (−0.999 − 0.0330i)20-s + (−0.995 − 0.0990i)22-s + (−0.959 + 0.282i)23-s + (0.137 − 0.990i)25-s + (0.411 − 0.911i)26-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (0.731 + 0.681i)4-s + (−0.754 + 0.656i)5-s + (−0.431 − 0.901i)8-s + (0.942 − 0.335i)10-s + (0.962 − 0.272i)11-s + (−0.0495 + 0.998i)13-s + (0.0715 + 0.997i)16-s + (−0.480 + 0.876i)17-s + (0.942 + 0.335i)19-s + (−0.999 − 0.0330i)20-s + (−0.995 − 0.0990i)22-s + (−0.959 + 0.282i)23-s + (0.137 − 0.990i)25-s + (0.411 − 0.911i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.411 - 0.911i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1845323538 - 0.2857576606i\)
\(L(\frac12)\) \(\approx\) \(0.1845323538 - 0.2857576606i\)
\(L(1)\) \(\approx\) \(0.5742094079 + 0.009888786512i\)
\(L(1)\) \(\approx\) \(0.5742094079 + 0.009888786512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.930 - 0.366i)T \)
5 \( 1 + (-0.754 + 0.656i)T \)
11 \( 1 + (0.962 - 0.272i)T \)
13 \( 1 + (-0.0495 + 0.998i)T \)
17 \( 1 + (-0.480 + 0.876i)T \)
19 \( 1 + (0.942 + 0.335i)T \)
23 \( 1 + (-0.959 + 0.282i)T \)
29 \( 1 + (0.277 - 0.960i)T \)
31 \( 1 + (0.592 - 0.805i)T \)
37 \( 1 + (-0.592 - 0.805i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (0.115 - 0.993i)T \)
47 \( 1 + (-0.989 - 0.142i)T \)
53 \( 1 + (0.556 + 0.831i)T \)
59 \( 1 + (-0.949 - 0.314i)T \)
61 \( 1 + (-0.685 + 0.728i)T \)
67 \( 1 + (0.329 - 0.944i)T \)
71 \( 1 + (-0.701 + 0.712i)T \)
73 \( 1 + (0.170 - 0.985i)T \)
79 \( 1 + (0.970 + 0.240i)T \)
83 \( 1 + (-0.724 + 0.689i)T \)
89 \( 1 + (-0.660 + 0.750i)T \)
97 \( 1 + (-0.863 + 0.504i)T \)
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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

−18.51260111717224336411968741837, −17.99055868127745036666226684517, −17.35813821862718064230268061928, −16.66634491538059901785172478193, −15.87957243768165506726295359993, −15.70187644757087721921718390911, −14.78129894361494762686172596420, −14.13231060599162701729054115494, −13.23342439037422363701430334852, −12.179854575520360433736847153348, −11.84207642497815137578251147723, −11.13452707037524336129413782712, −10.22057188774833366734340317425, −9.60720255116833830881103782326, −8.843827938285184526793125441878, −8.33610409990119239006639450760, −7.59240645426979814868776625615, −6.94201899993959285527049887993, −6.25218475749663138902315153990, −5.12128559544064872662115079469, −4.789214817526575765194490272, −3.52204336809175851668609203293, −2.81865122932875421664202200266, −1.55609188426082934010851012105, −0.94719231867436852138868443556, 0.15633406332897433149208266689, 1.43008324695098075269531299219, 2.11010183456104963843900685157, 3.10088914406594465555970750737, 3.89490392263585250544875344378, 4.26168109915464114694541285521, 5.878279361263244900083035620113, 6.52914167613983658866837478063, 7.119746112992857275152521486817, 7.92650716345648095239171965831, 8.47106356608139336173121293176, 9.312543020681718243044221117, 9.920685868668058621943548930096, 10.70801028358035771625716035300, 11.39285210070683524317695773592, 11.928470936536653161888369321401, 12.291930854015642454340928947875, 13.63105316751657121783566662407, 14.10499741592689909954307276842, 15.19069108221446004063897960500, 15.508835071735612740236173773425, 16.48240616798105162680728456376, 16.84553823638287596398353278102, 17.73940810446966682339573817747, 18.31103339102002208633818945009

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