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Properties

Label 1-4033-4033.1106-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.744 + 0.667i$
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 + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5990901559 + 1.565695653i\)
\(L(\frac12)\) \(\approx\) \(0.5990901559 + 1.565695653i\)
\(L(1)\) \(\approx\) \(1.411112560 + 0.3382256650i\)
\(L(1)\) \(\approx\) \(1.411112560 + 0.3382256650i\)

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 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−17.97330069976598906036501904192, −17.203566136999657182424581689660, −16.890799999944189944859015106328, −16.47382618066163619512772350457, −15.32240514412143582953915640478, −14.54309299175956527070032305469, −14.06024294611772892165255225411, −13.53683826090795014662581643753, −12.77187365910043742076302259630, −12.25016417657477951151620994712, −11.80240418711176135123823180320, −10.88478761124247609187604353951, −9.50499892082996953884494336788, −9.14887207542506087009631924469, −8.148172081202125032777254249487, −7.3723767082122831068812735072, −7.21024770850818895501562502702, −6.30949020660552904659640162573, −5.38915219037081204723342336897, −4.84828116797364488798927693050, −4.01160676280153065428299427754, −3.28311714016149634664750505989, −2.08305324806766418629304208611, −1.522863777145352621563578091417, −0.30791412959666226364013509060, 1.43618819853228159353640297152, 2.511090616382317272592538304265, 2.88761206813933813465778395792, 3.59655739569039628364724770467, 4.39397217145732665313676605525, 5.282807272350255298157639736930, 5.77142109602347797698377981343, 6.48178126186447489169869138068, 7.447481178935185384891109439040, 8.59084369861392582774046202361, 9.36591636725919456965772027269, 9.72994814629372588327029803481, 10.68880393784958919525517671519, 11.26147512824441412677374083143, 11.51945933826973245033694588244, 12.62002592313334703267131859505, 13.224884484814545974457739835912, 14.34448091596958824630140374711, 14.58833353413648089259668474751, 15.08190783204620800536063235159, 15.62913049113243674766997647744, 16.582952770883277018932450102646, 17.38640774929700308217112588699, 18.17440781517453745667668859030, 19.07069337107192555553686077501

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