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Properties

Label 1-4033-4033.2238-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.363 - 0.931i$
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  − 2-s + (0.835 + 0.549i)3-s + 4-s + (0.802 − 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.835 + 0.549i)3-s + 4-s + (0.802 − 0.597i)5-s + (−0.835 − 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (0.802 + 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.998 − 0.0581i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.586000868 - 1.084088324i\)
\(L(\frac12)\) \(\approx\) \(1.586000868 - 1.084088324i\)
\(L(1)\) \(\approx\) \(1.137601201 - 0.1893135019i\)
\(L(1)\) \(\approx\) \(1.137601201 - 0.1893135019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (0.802 - 0.597i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.802 + 0.597i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (0.957 + 0.286i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (0.448 + 0.893i)T \)
59 \( 1 + (0.0581 + 0.998i)T \)
61 \( 1 + (0.727 + 0.686i)T \)
67 \( 1 + (-0.549 - 0.835i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (0.0581 + 0.998i)T \)
83 \( 1 + (-0.973 - 0.230i)T \)
89 \( 1 + (-0.116 - 0.993i)T \)
97 \( 1 + (-0.957 + 0.286i)T \)
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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

−18.85245514024578831866525083463, −17.909421291984652635615078723567, −17.48614375944114246076903823192, −16.82484392908321025194442661673, −15.899685688344834996419806472441, −14.93515038740516472946226262477, −14.675878860040403160717570799224, −14.102884254926168839165889828206, −13.053905669596695339908232857791, −12.41838762020568083154852394222, −11.51779363763631863265410266595, −11.12429499049855092706264648217, −9.97741503723011905219132070210, −9.4602977358500073895751297710, −8.90085941106679246787184379192, −8.31603800773824949242708914357, −7.52147961516092154026872314930, −6.729238306333782223057167044778, −6.21298495015537663686472405549, −5.56842588535738065493872704033, −4.0825955228480879750040625512, −3.13607739368191246929897630311, −2.470840315530448548666118321867, −1.74206016080242886388178349210, −1.33896139496400717045725884294, 0.63302169917191485358697388864, 1.45330743686252582264790900643, 2.38369724951651132407749722773, 2.85601536513770865648651431555, 4.14519614262008543241770749177, 4.71784137644022115028973951853, 5.56594924636423108628299061477, 6.77750877731429449735707761639, 7.26186673659825866794087237237, 7.98203151362699174780766618695, 8.98819101820851390982423526978, 9.10549509723039931729323366881, 9.98508404888491689816369179502, 10.42880296683729220530578971384, 11.155553992945588041689358209487, 12.085265464234067265196474113433, 13.00578135460357420453916284078, 13.58798880226441414927540052478, 14.433359690199571307770136884223, 14.98382440035003112255569690978, 15.683640118804384567863672043477, 16.51357895071710437024681776290, 17.06675975176150726020370913369, 17.54755580003324466001705847988, 18.13815941056127887563266421936

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