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Properties

Label 1-4033-4033.618-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.745 - 0.666i$
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  + (0.766 − 0.642i)2-s + (−0.686 − 0.727i)3-s + (0.173 − 0.984i)4-s + (0.893 + 0.448i)5-s + (−0.993 − 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.835 + 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (−0.286 − 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.686 − 0.727i)3-s + (0.173 − 0.984i)4-s + (0.893 + 0.448i)5-s + (−0.993 − 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.835 + 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (−0.286 − 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (618, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.745 - 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5796982225 - 1.517160724i\)
\(L(\frac12)\) \(\approx\) \(0.5796982225 - 1.517160724i\)
\(L(1)\) \(\approx\) \(1.035397948 - 0.6519718644i\)
\(L(1)\) \(\approx\) \(1.035397948 - 0.6519718644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.686 - 0.727i)T \)
31 \( 1 + (-0.0581 + 0.998i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.893 - 0.448i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (0.597 - 0.802i)T \)
97 \( 1 + (0.893 + 0.448i)T \)
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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

−18.48875937788548373236782404789, −17.48784950132732269761123940622, −17.139880523391718560840464810439, −16.70597718499501629132155220843, −16.215786705771776204506604160767, −15.22366368597929807172365763799, −14.7747726111583676139538952915, −14.058601490493763726055959735052, −13.13557204362790130228685676140, −12.72487641301693152809414298676, −12.16805586069774086077800057627, −11.06767287311761424194527636435, −10.60243028753975491452307342765, −9.577356705997088894281119976125, −9.175502837801436216293049012500, −8.36372968439341588996673326915, −7.12359326645214275443276700284, −6.54431097803427288903939545287, −6.01788864058251396841969857364, −5.39034795670508623026864167038, −4.41627040176744091398838213770, −4.14683952370436348551468475813, −3.138514947635669255053176767632, −2.28171488157558432133422388973, −0.89783003905058858510856290672, 0.42170519746975728637840822333, 1.77672480208483940019014836646, 2.09613953057896665643031725373, 2.8643204716409452936742467796, 3.87013122908402892810377203480, 4.89236447684309687827718140454, 5.416466930953656638207235380161, 6.33694595531901236252498216744, 6.69620054767939423309853922025, 7.11720522277074824857394735195, 8.761761909742382040634522600862, 9.45115171819992251317249460549, 9.95441888803243759757963396774, 10.908468381907940036738427963803, 11.42895032028013000009276350562, 12.08604973140291900648238749654, 12.85675943139544045651326437539, 13.175080935453514666946921555593, 13.959831729633822458566325684961, 14.591414189977870933998345706786, 15.31305295109010975370351220714, 16.139129335958325168331961770590, 17.228901991962739799141332298223, 17.34551354610106684170500508818, 18.51734135594877255416556896027

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