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Properties

Label 1-4033-4033.1274-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.599 + 0.800i$
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 + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.541495881 + 1.271015135i\)
\(L(\frac12)\) \(\approx\) \(2.541495881 + 1.271015135i\)
\(L(1)\) \(\approx\) \(1.650527768 + 0.05781339645i\)
\(L(1)\) \(\approx\) \(1.650527768 + 0.05781339645i\)

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 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.766 + 0.642i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−18.25982788568126038001949539980, −17.53670587362560882653996934509, −16.94489789212521987074325551182, −16.58273991969732155143968452702, −15.58570109110497891687212412551, −14.60066693921277449420871509079, −14.36562667642309401541990238435, −13.7151752551227727159200710115, −13.14438767760560935077338028440, −12.64800786116978813786948761916, −11.6122119244627035214622827442, −10.83361484688795298990345627621, −9.52872551644819958485075396016, −9.11749156785454764216316863781, −8.59946545155212745321944942201, −7.95058027011733471855489850473, −6.971229936765439322816151803387, −6.64255435339918308536968342574, −5.812229970424529408492301148415, −4.90259206811492002177750953286, −4.143000167118311030693371049, −3.59677713437522882661917133260, −2.22697320506091768123694147519, −1.53710213135263663678743916214, −0.649037201104898269450605926935, 1.33612976791862621237476278899, 2.264417469405228959058777929422, 2.36859325913824890281143085214, 3.48972086039620200854122572605, 4.085385378384129552899118521883, 5.03501634525641033661435064626, 5.475412462535977001612086202304, 6.5404486383423801979190399105, 7.541007229249665081494286302364, 8.641202613494758893719995501973, 8.88456891482436822398877277065, 9.642530263923861462261765372061, 10.37510412021101657513760292094, 10.926420754303944225722499610642, 11.424271114107794115074644757963, 12.5299493161061629336410544301, 13.20326835769520684791124778206, 13.66606776553157410419342159788, 14.71324732146441988453889390686, 14.9648649223106550276658585108, 15.22939511192475738885353364664, 16.76529185508868613740588580398, 17.44934514003137676586289454495, 17.99425052535805428236388645381, 18.65792719200989681440836874111

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