↑

Properties

Label 1-4033-4033.2981-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.325 + 0.945i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + 12-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7194399652 + 0.5129909334i\)
\(L(\frac12)\) \(\approx\) \(0.7194399652 + 0.5129909334i\)
\(L(1)\) \(\approx\) \(0.6361072860 + 0.2873887720i\)
\(L(1)\) \(\approx\) \(0.6361072860 + 0.2873887720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.335871429257830767794000191910, −17.68220454926359476501778482869, −17.209425395360854291947229975177, −16.633252160413752978507646048545, −15.89308246353208317112573528163, −14.603685587817413669598442427973, −14.06966573456201635326016261131, −13.26747157506965701943888856398, −12.76685456048719714587271574043, −12.09782130401805881855079649030, −11.6159502352402621732018859490, −10.57559611346930539576121819277, −10.082779447652390223110649200151, −9.69277190910283444690565863761, −9.0455501902893433154451045801, −7.53815529271004859442245003451, −7.24648714133628622430885965796, −6.29179319149422721953340232900, −5.539492614642618424339460861523, −4.871078772499321329194665796388, −3.9581195290677898103470390457, −3.14984324244766869513650905760, −2.00135235955306849327994278254, −1.758130869627395114261506907202, −0.51453003988657609400396221677, 0.6168588980048733947496947691, 1.432805149013183313770827300774, 2.94868456907271334672088174969, 3.70502522144923691678112228057, 4.77436379970092370018601465173, 5.56488345064880309042094533518, 5.78260853203866545522799522596, 6.37468258517035512259885335759, 7.29980238235453031608239837847, 8.02979008486097291147526573806, 9.12518220924442030607977487407, 9.6750812825469256733217804977, 9.92605860038042313190611902321, 10.829090280143448192251158972913, 11.85918771874638967004459797248, 12.69171036941261635595693153877, 13.02271422046347983750839053710, 14.008492698945664840873228040899, 14.531248584859753898013639856372, 15.44772435569927276063965051275, 16.213712624811065614103227428012, 16.55108637872619028754604639085, 16.9294055197185605563988785725, 17.94227135210238244298105199996, 18.24015571482486185088560979616

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