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Properties

Label 1-4033-4033.206-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.340 - 0.940i$
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 + (0.396 − 0.918i)3-s + (−0.939 + 0.342i)4-s + (0.686 − 0.727i)5-s + (−0.973 − 0.230i)6-s + (0.973 + 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.0581 + 0.998i)12-s + (0.286 + 0.957i)13-s + (0.0581 − 0.998i)14-s + (−0.396 − 0.918i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.396 − 0.918i)3-s + (−0.939 + 0.342i)4-s + (0.686 − 0.727i)5-s + (−0.973 − 0.230i)6-s + (0.973 + 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.0581 + 0.998i)12-s + (0.286 + 0.957i)13-s + (0.0581 − 0.998i)14-s + (−0.396 − 0.918i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198927204 - 1.709408450i\)
\(L(\frac12)\) \(\approx\) \(1.198927204 - 1.709408450i\)
\(L(1)\) \(\approx\) \(0.9228016827 - 0.8417796718i\)
\(L(1)\) \(\approx\) \(0.9228016827 - 0.8417796718i\)

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 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.686 - 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (-0.0581 + 0.998i)T \)
59 \( 1 + (-0.396 - 0.918i)T \)
61 \( 1 + (0.835 + 0.549i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (0.835 - 0.549i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (0.993 - 0.116i)T \)
97 \( 1 + (-0.396 + 0.918i)T \)
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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

−18.28319279319110706595478184328, −17.93410279274365135195318865618, −17.30341954398764119165748519484, −16.645485303869924753611768826486, −15.6865956544677752290963587167, −15.39504217736595516262375778266, −14.66334669741625267544016019791, −14.13813576293618999980991726332, −13.51617555638885135065566645980, −13.000883761478411816778962051299, −11.488955306132893297458712862331, −10.77509338699631653620431835419, −10.23847168239539195517033700712, −9.8206911094787554195351471628, −8.72616298448629211996669179488, −8.1419060495936593112411387487, −7.82839783792093214119034629646, −6.63559756936177698106667290832, −5.948578266331747682572210525965, −5.247401309173651435935186642335, −4.69855756970946090884022107868, −3.77890241700610000544477018867, −2.93448177553325123575555334031, −2.09334235934501395344791581178, −0.76083434888485275453947865462, 0.8587793892722368607221907976, 1.564667605666293365816591066042, 2.12541689922478912628344121085, 2.728479986985453851494244546378, 3.86302353786825923969792324032, 4.79611603668827856492540763102, 5.28692678097097485845574227212, 6.21022913775261173898032000090, 7.34481434584343205408832199555, 8.01208196045407851785435138141, 8.5423419021971304338835036565, 9.29108774971824379924571606945, 9.817808985047884204525249711017, 10.74147808937729545661528255416, 11.6894538122076755943386496291, 12.06148187118600261597520411670, 12.652256819090142724759895729948, 13.49563929189920029643531757089, 14.08279443237421159011484825768, 14.214881842396690492356121562283, 15.55259674304842528149812101370, 16.40593696103233119335308585213, 17.39136507330926911077459896459, 17.745409009082480323699190572271, 18.34712302070214906532933969323

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