Properties

Label 1-4033-4033.1021-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.434 - 0.900i$
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 + (−0.396 + 0.918i)3-s + (−0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.973 − 0.230i)6-s + (0.973 + 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.549 − 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.0581 − 0.998i)12-s + (−0.286 − 0.957i)13-s + (−0.0581 + 0.998i)14-s + (0.918 − 0.396i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.396 + 0.918i)3-s + (−0.939 + 0.342i)4-s + (−0.727 − 0.686i)5-s + (−0.973 − 0.230i)6-s + (0.973 + 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.549 − 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.0581 − 0.998i)12-s + (−0.286 − 0.957i)13-s + (−0.0581 + 0.998i)14-s + (0.918 − 0.396i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.434 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1609169688 - 0.2563617010i\)
\(L(\frac12)\) \(\approx\) \(0.1609169688 - 0.2563617010i\)
\(L(1)\) \(\approx\) \(0.6407526091 + 0.2526454609i\)
\(L(1)\) \(\approx\) \(0.6407526091 + 0.2526454609i\)

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 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.549 - 0.835i)T \)
13 \( 1 + (-0.286 - 0.957i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (0.116 - 0.993i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.957 - 0.286i)T \)
53 \( 1 + (-0.998 - 0.0581i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.448 + 0.893i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (-0.835 + 0.549i)T \)
83 \( 1 + (0.686 - 0.727i)T \)
89 \( 1 + (0.116 + 0.993i)T \)
97 \( 1 + (0.918 + 0.396i)T \)
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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

−18.7246408116192834857955468760, −18.07218708896476196782094877618, −17.750981107931327630612481182555, −16.92218546240158560677966492624, −16.02810533710779727366864477406, −14.81421000864415013342750187508, −14.47737319330486334260273356442, −13.99071798573292756852282395053, −12.88183577277474885933024959690, −12.45960034139382429266346431420, −11.787790865376358529846099868377, −11.310913387850632160098491090582, −10.44826982970681040051836333225, −10.26875589178694500530925333358, −8.83123324788760725356261246589, −8.120110136236149674980088369287, −7.70628652274920212167806671536, −6.704019905412340937432225548985, −6.10436389488440510505690102011, −4.847868450570599011130379275756, −4.56373525782498449743372596159, −3.603132033211103706683952300393, −2.60566076315357883666855121187, −1.86351816063306974769775104476, −1.396023332212582474206272357015, 0.11666034915138434180162063666, 0.78137091395619263529859693738, 2.57894697958719731409620303141, 3.4864616664316168951235535233, 4.30441202209525781055414913881, 4.841067862696675640597547973105, 5.43893382027214258911418937261, 5.92293109937249987237024182824, 7.1344174992184293379202525842, 7.92118232461907709994673065364, 8.39958610870115765467774244635, 8.99923371507015150331139314406, 9.80732266587917273111615698135, 10.66238635439009042525031433921, 11.58479198910545640546803326480, 11.85849171307282150560072510833, 12.92843165653946206479229196689, 13.581878499305627715757635018192, 14.39272616664579787162994325847, 15.15316226988740603543699953134, 15.7212388636043352550997681500, 15.8835109844093429234371777650, 16.83958945075799893663249998502, 17.41899151346748965495936380976, 17.86032416596274008638684219561

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