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Properties

Label 1-4033-4033.2028-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

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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: $\chi_{4033} (2028, \cdot )$
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 \( 1 + (0.939 - 0.342i)T \)
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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

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

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