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Properties

Label 1-4033-4033.3548-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.986 - 0.166i$
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 + (0.973 − 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.893 + 0.448i)5-s + (0.0581 + 0.998i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.993 − 0.116i)12-s + (0.0581 + 0.998i)13-s + (−0.396 − 0.918i)14-s + (0.973 + 0.230i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.973 − 0.230i)3-s + (−0.939 − 0.342i)4-s + (0.893 + 0.448i)5-s + (0.0581 + 0.998i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.597 + 0.802i)10-s + (−0.597 − 0.802i)11-s + (−0.993 − 0.116i)12-s + (0.0581 + 0.998i)13-s + (−0.396 − 0.918i)14-s + (0.973 + 0.230i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.986 - 0.166i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.986 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1122868889 + 1.340233087i\)
\(L(\frac12)\) \(\approx\) \(-0.1122868889 + 1.340233087i\)
\(L(1)\) \(\approx\) \(0.9120430915 + 0.6853804771i\)
\(L(1)\) \(\approx\) \(0.9120430915 + 0.6853804771i\)

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 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.893 + 0.448i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.597 - 0.802i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.893 + 0.448i)T \)
53 \( 1 + (0.993 + 0.116i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (-0.597 - 0.802i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (-0.396 + 0.918i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (0.973 - 0.230i)T \)
97 \( 1 + (-0.286 + 0.957i)T \)
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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

−18.16371896031479040615385761748, −17.74076210406522654104415810425, −16.70947472719468906560310354058, −16.27594642566791221486723821503, −15.248951528454625783829896035763, −14.52507767107145518462696194324, −13.75148142679831590716268701889, −13.22280164129374551927706916690, −12.75946501531403797308341876687, −12.29247499189154386909515720603, −10.82531864763809585606752299628, −10.28290186872743178664082696133, −9.98785386388104869279565718104, −9.13858060333525950124882720361, −8.73550341588742149601583408199, −7.7974715906041322256579832980, −7.131897354600599169461825519863, −6.02490558545593212897044412541, −4.9464485772794204097749159295, −4.49270983634569655607747264783, −3.589792568450934687431862807685, −2.676999933461970971409818820405, −2.39394999358329720033818917897, −1.38841144685324911166636234576, −0.32613214053846124502743405651, 1.361359744797482946251083692955, 2.15092234877084800557327083842, 3.03086253497871549980389616490, 3.779431448197381600607786508116, 4.67089606665571899032141455301, 5.91568315295697468482053476798, 6.12704552852895111790658834116, 6.85466942246727849851831708501, 7.6439949672519306227646598559, 8.50870669995629705120812224983, 8.97678736072725953675411561374, 9.589355339567888542724690632503, 10.21650952032953290230561097691, 11.00442816627747100960164495417, 12.39322665511734948772613283081, 13.05686800359408786998696589942, 13.49460497100189264477721707467, 14.13158472506579287612010455416, 14.73475976838797228094504393917, 15.395949211819477593674935994738, 16.01502137799606719592451579581, 16.68666619757252554018894456165, 17.53823615680860338415826298526, 18.167834205795212420530383359536, 18.85257421893781658624734874708

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