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Properties

Label 1-4033-4033.4-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.275 - 0.961i$
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 + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s − 5-s + (−0.939 − 0.342i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.173 + 0.984i)14-s + (0.5 − 0.866i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s − 5-s + (−0.939 − 0.342i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.173 + 0.984i)14-s + (0.5 − 0.866i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.275 - 0.961i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4664214878 + 0.6186352475i\)
\(L(\frac12)\) \(\approx\) \(-0.4664214878 + 0.6186352475i\)
\(L(1)\) \(\approx\) \(0.4105808724 + 0.6173239496i\)
\(L(1)\) \(\approx\) \(0.4105808724 + 0.6173239496i\)

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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
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

−18.34323977640998420146862675, −17.39150920678411720186537294278, −17.0750456852708237174421928429, −15.79433118773634800738510552809, −15.25825407812539726516601185885, −14.33718650774850231496363338701, −13.63635383760296215097942220996, −13.16884612333432812966476862953, −12.27562583633339317489465835652, −11.76168448502762114896867935822, −11.19765751177944329467922413622, −10.87923625088969252778699107057, −9.92676481347059214460271327500, −8.72614486980956108177992454447, −8.206059844021809833181852330463, −7.73266190755364272833711099248, −6.79973933803011187218051953034, −5.677580362022846775928676876352, −5.184967925083075249134676737011, −4.43878288921164909080812354214, −3.53234825359831931908109838400, −2.697819387156777095672184061212, −1.9513996008097464371655659191, −0.90608840427628179770913843774, −0.32059964440718555097313474134, 1.05497133552871374517520962188, 2.498082431191353529811937849233, 3.63161434368117885383370232017, 4.38688571648269987514161488988, 4.70747190078535638575274366529, 5.2501375454843966016852293185, 6.38736455399596422228716749211, 7.05376710469800157979544755901, 7.66319449958453713627903243555, 8.554394174481651568875264035, 9.00462044385402168176864810258, 9.95453418990205690591345868488, 10.609999540600492092061675247635, 11.57188200514390767143552261916, 12.0616908296355721637646538645, 12.5864418121538292449262499762, 13.904437584536429063262290324829, 14.50799319290122575243265393419, 14.8800503410806229861725189941, 15.719080943652222411581199324577, 16.062210707557565395338120648674, 16.72471607744556657109573948965, 17.52827615888260905822392007874, 18.00167336324358071726071652928, 18.59768055910451460702829213708

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