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Properties

Label 1-4033-4033.529-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.796 + 0.604i$
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  + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (0.173 − 0.984i)12-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.939 + 0.342i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (0.173 − 0.984i)12-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (0.939 + 0.342i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5747258454 + 0.1933846602i\)
\(L(\frac12)\) \(\approx\) \(0.5747258454 + 0.1933846602i\)
\(L(1)\) \(\approx\) \(0.5595534417 - 0.1381281421i\)
\(L(1)\) \(\approx\) \(0.5595534417 - 0.1381281421i\)

Euler product

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

−17.965720212378583208319685963930, −17.78941669267526020281402440627, −17.296966454958735755556473403014, −16.324556526261359276225730165159, −15.69626731160200658775540591043, −15.25602522640570303604599056384, −14.53143925490751754850506302955, −13.60378336550626065918170649588, −13.1527086458857398715686057950, −12.0049428259662847391683551176, −11.43299729401514704124287474036, −10.670365417055930504433280861717, −10.318685633953401703729028096421, −9.473449397746530062238637615461, −8.14615854821430542691028274864, −7.798292222793707471965515135001, −7.33479729426690825968788400553, −6.48852249983569588005636465668, −5.93588173505072912201802919139, −4.99610471635303267305237911525, −4.385163159399111538542803408917, −3.65724514172154035053891758722, −2.07214340307370853765974407955, −1.2981559232138882882916565883, −0.326521371900525513647119630629, 0.91186049496493486988686422053, 1.36336659867398399593293857158, 2.6610232903974222141771270216, 3.509837734299015396811430948662, 4.36974775588317898110164937461, 4.84525476338756211857258304933, 5.597419249968672178509012845290, 6.6038451819490998289334661822, 7.594634297187151662133713500390, 8.288827439422199269913521837273, 8.98935255354125905980767903584, 9.39390363540402540062142056892, 10.58824925484618935398244516897, 11.09194454546412748504436436981, 11.62597657409678089809335583361, 12.0246960844326299494083578381, 12.71337805247893091388377017591, 13.613710554204723884199861997967, 14.30240704272462357033878812856, 15.56836769940404792832118138939, 16.075666753150337849833513231622, 16.434066160974976620913671688144, 17.24869316445847552829265718994, 18.09254092431024796391515894446, 18.45140590125507667657536345042

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