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Properties

Label 1-4033-4033.2637-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.690 + 0.723i$
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 + (−0.686 − 0.727i)3-s + (−0.939 + 0.342i)4-s + (0.893 + 0.448i)5-s + (0.597 − 0.802i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (−0.835 + 0.549i)13-s + (−0.686 − 0.727i)14-s + (−0.286 − 0.957i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 + 0.342i)4-s + (0.893 + 0.448i)5-s + (0.597 − 0.802i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (0.893 + 0.448i)12-s + (−0.835 + 0.549i)13-s + (−0.686 − 0.727i)14-s + (−0.286 − 0.957i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.690 + 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077605743 + 0.4610673386i\)
\(L(\frac12)\) \(\approx\) \(1.077605743 + 0.4610673386i\)
\(L(1)\) \(\approx\) \(0.8007154304 + 0.3280294569i\)
\(L(1)\) \(\approx\) \(0.8007154304 + 0.3280294569i\)

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 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.973 - 0.230i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (-0.835 + 0.549i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (-0.0581 + 0.998i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (-0.993 - 0.116i)T \)
97 \( 1 + (-0.835 + 0.549i)T \)
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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

−18.16233136795141134979657762852, −17.71407913582606475887997559771, −17.08837173880782358825694189604, −16.64404205548635748331246245685, −15.63164268090270966668221403506, −14.89195495117898844565276777192, −14.264387478099586964228211220277, −13.27288673211538130948805096382, −12.77622258777765117520748285040, −12.33971728422065245731614287805, −11.520227335077729167698405208135, −10.48164019281801356989660130943, −10.13066261148560071169825422721, −9.80186500786002362924294565117, −9.07354771176590277982781668822, −8.1884742147668260531550972559, −6.86290313268902365396554195407, −6.29198001018342536458814090358, −5.22828339855836773817117586940, −4.90079307040657150503778517177, −4.260293848448906543582610876715, −3.10239800467779551446905756785, −2.71813536316550607909520767427, −1.42360691870948233481338862443, −0.67496200671275865569975689059, 0.54357298030200585485278666197, 1.7134681698746395571618299347, 2.83580971824171279977291254446, 3.338101646411187148511351597549, 4.80902663842418220088331867177, 5.40988007853481966963508402158, 5.94350609191353023564198890505, 6.55459073118152666073015965194, 7.07300324474925391977844877723, 7.91775952554893816632838104809, 8.65142833522425376866747680337, 9.77769540148653739130865580829, 9.857244910102676319069134875030, 11.04984859969359293437516124640, 12.02794437368172301971835552189, 12.419555398945822682340687806423, 13.34606739764094935448683774641, 13.76022252828477949809386242008, 14.29231269264581007411160470146, 15.20879318029952260212801039018, 16.04286339459832134302928032126, 16.62103204511329384527986667202, 17.06862364843258686587557519535, 17.79429262214000770470369269159, 18.50618248675131713971083403240

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