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Properties

Label 1-4033-4033.1728-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.310 - 0.950i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1728, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5004898922 + 0.6900133379i\)
\(L(\frac12)\) \(\approx\) \(-0.5004898922 + 0.6900133379i\)
\(L(1)\) \(\approx\) \(0.6437979440 + 0.7358097697i\)
\(L(1)\) \(\approx\) \(0.6437979440 + 0.7358097697i\)

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

−17.87393943015521240363232296283, −17.569907957492855961984170559759, −16.7146963846522882469467601184, −16.26678999416901183723503876924, −14.91317422946398969188656756360, −14.47559241163892404463731827032, −13.701450597475629676120751231178, −13.01952937039103598693838122984, −12.43128535596463092098086210736, −11.99776054247804040906902277137, −11.155158831169318435627863368330, −10.43837194306969918248649123864, −10.035762726237989132076169632092, −9.20542326919545332977862731853, −8.371890698997027559985208426406, −7.09303615797844352883146590494, −6.814440326370445286215064004458, −5.6206743751616514672963711301, −4.97662620915084351321035133166, −4.730280785152045137109283995717, −3.92056010481532614407372062776, −2.42304544143806541636937312351, −1.89767090170601810538005275731, −1.23774288145527675044743166061, −0.22630256533875695082655278275, 1.33899841313042258473585882251, 2.50877079768018697121567027506, 3.2934395316570715782675200135, 4.22926526963254563489120105273, 5.07576603694998114120113354907, 5.69337365967249366761173047682, 5.99903827857237033363892843408, 6.762976162150581305621566865699, 7.72803729460049520458191489162, 8.28116174575831147195536154557, 9.32312880120849695617579033711, 9.88697033301924451747052114921, 10.94669815935688600038349032203, 11.25005656113045050369887851864, 12.226795574238459853685151989282, 12.87375183051621352864804826711, 13.470339129323533938669680575279, 14.51990233948731190155664495550, 14.88712831172609172723370657716, 15.3741966819694446242854033359, 16.28909228068391870750719259515, 17.018228384156658796776123023094, 17.416678109809861233392896435014, 17.948683195896716456283819819997, 18.65658391763769384554056948944

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