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Properties

Label 1-2001-2001.65-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.962 - 0.271i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (−0.262 + 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.768 − 0.639i)5-s + (0.818 + 0.574i)7-s + (0.714 − 0.699i)8-s + (0.818 − 0.574i)10-s + (−0.591 + 0.806i)11-s + (−0.0611 + 0.998i)13-s + (−0.768 + 0.639i)14-s + (0.488 + 0.872i)16-s + (−0.142 + 0.989i)17-s + (0.862 + 0.505i)19-s + (0.339 + 0.940i)20-s + (−0.623 − 0.781i)22-s + (0.182 + 0.983i)25-s + (−0.947 − 0.320i)26-s + ⋯
L(s)  = 1  + (−0.262 + 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.768 − 0.639i)5-s + (0.818 + 0.574i)7-s + (0.714 − 0.699i)8-s + (0.818 − 0.574i)10-s + (−0.591 + 0.806i)11-s + (−0.0611 + 0.998i)13-s + (−0.768 + 0.639i)14-s + (0.488 + 0.872i)16-s + (−0.142 + 0.989i)17-s + (0.862 + 0.505i)19-s + (0.339 + 0.940i)20-s + (−0.623 − 0.781i)22-s + (0.182 + 0.983i)25-s + (−0.947 − 0.320i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.962 - 0.271i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09141883013 + 0.6616389835i\)
\(L(\frac12)\) \(\approx\) \(-0.09141883013 + 0.6616389835i\)
\(L(1)\) \(\approx\) \(0.5927932517 + 0.4236624348i\)
\(L(1)\) \(\approx\) \(0.5927932517 + 0.4236624348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.262 + 0.965i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
7 \( 1 + (0.818 + 0.574i)T \)
11 \( 1 + (-0.591 + 0.806i)T \)
13 \( 1 + (-0.0611 + 0.998i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.862 + 0.505i)T \)
31 \( 1 + (-0.992 - 0.122i)T \)
37 \( 1 + (-0.794 - 0.607i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (0.992 - 0.122i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.742 - 0.670i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.101 - 0.994i)T \)
67 \( 1 + (0.591 + 0.806i)T \)
71 \( 1 + (-0.882 - 0.470i)T \)
73 \( 1 + (-0.301 + 0.953i)T \)
79 \( 1 + (-0.488 + 0.872i)T \)
83 \( 1 + (-0.452 - 0.891i)T \)
89 \( 1 + (-0.523 + 0.852i)T \)
97 \( 1 + (-0.970 - 0.242i)T \)
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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

−19.71827749454142088377647614582, −18.78563935558384848868676256277, −18.16587787952608918393540205479, −17.79496291956218753416059517920, −16.740479590825747175686667265300, −15.967990553603717390661008628465, −15.11283094028782930437540856489, −14.217652621757883882855610655476, −13.63577110125750050567960803139, −12.9105940731879008231941253069, −11.7392119192858255632505133984, −11.497422950339089022813579998982, −10.60708599655690788562608542282, −10.28889925654768022620410391225, −9.08920952190170071412861350615, −8.27225289848307930020861488360, −7.63974191226767028560480358530, −7.08075407812765290548916306287, −5.49882813659552449111101726758, −4.87245466576379188464994938327, −3.88201681880691968200095950642, −3.126184671528565043118388172471, −2.53693156599348380560202936771, −1.16916861224772376016130585923, −0.29466780652292736756281946822, 1.29637450921208840553347840081, 2.08615211612850527818586038646, 3.79211594741390226310325565320, 4.37421989339170036814222979946, 5.236615005978228205576955481704, 5.73099226946921211221375680655, 7.05140697689643306364946046367, 7.52674222855354169519372683556, 8.34752605916209587024935476774, 8.87569947098669957349371704078, 9.65022245198226339511156597763, 10.644067014638295285809738563073, 11.52716572428571115477744028360, 12.395320559316554735787325507011, 12.920075417825497480584971559, 14.08917083331586012079795504941, 14.6069709609068026691214391843, 15.43207250590540111833522773354, 15.86556107864900338032076247469, 16.65262848842696716805704849752, 17.40417849469616996124996387651, 18.01970079404225760955634691515, 18.84668034606903755417962639588, 19.345679298004853093804892313601, 20.36265285023204862931577575096

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