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Properties

Label 1-4033-4033.1027-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.979 + 0.202i$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)6-s + (−0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s − 10-s − 11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + 14-s + (0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)6-s + (−0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s − 10-s − 11-s + (0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + 14-s + (0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.979 + 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001770562759 + 0.01729502385i\)
\(L(\frac12)\) \(\approx\) \(0.001770562759 + 0.01729502385i\)
\(L(1)\) \(\approx\) \(0.7598348561 - 0.03362083560i\)
\(L(1)\) \(\approx\) \(0.7598348561 - 0.03362083560i\)

Euler product

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

−18.32141775198305161947182620911, −17.43987185881173473478958056853, −16.814248630075110699421064491892, −16.15475340459209963408958049194, −15.52526315751349897006905489684, −15.126877985024712552620965942651, −13.860782622021564249337933026847, −13.33824859450754039729606806908, −12.72427978476119767323265046001, −12.02153650857324543809019437057, −10.844724928302118262696828783570, −10.20851881537694915867959596136, −9.85710066445065842513024012794, −9.34129859023410285093680704474, −8.45508828962076498475231658559, −8.02375174826335127401836469359, −7.204848934907825875131370901045, −6.02217158502921776458604179788, −5.62908466511653241609922597698, −4.506748334094707457027887824286, −3.43815670402587658546982198907, −2.79210689225867993927080576978, −2.35096407333539489417294383313, −1.3352446529464959070161033417, −0.0053838533383787264841518547, 1.34354799712145463304091289077, 1.9321775075167279142022588309, 2.778541928175389863217837932349, 3.309947705257994069555994646220, 4.67483881088503089698015634503, 5.807034261662448563255108382996, 6.58106083986168905423393598102, 6.70496187003634122073070920913, 7.63331527687604449946970309103, 8.5140977410847456437076903806, 8.89588556518948184401716004028, 9.85445491851595330505675858199, 10.21774046379496592917834248123, 10.83579577053719492702584513223, 12.064916619386549979099902106759, 12.70873539264489994702037192024, 13.476648131851561297698480906334, 14.043086268083053680875079383567, 14.72190906165840955679486908379, 15.35761696114097868736928185736, 16.30594685865363894328075898059, 16.74082649350690674100170075803, 17.58624266050439849615279504307, 18.22383687829263120429552412658, 18.78741228555201985722110467247

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