Properties

Label 1-4033-4033.2090-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.563 - 0.826i$
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  + (0.939 + 0.342i)2-s + (−0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.642 − 0.766i)5-s + (−0.939 + 0.342i)6-s + (0.766 − 0.642i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s − 12-s + (−0.173 − 0.984i)13-s + (0.939 − 0.342i)14-s + (0.984 + 0.173i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.642 − 0.766i)5-s + (−0.939 + 0.342i)6-s + (0.766 − 0.642i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s − 12-s + (−0.173 − 0.984i)13-s + (0.939 − 0.342i)14-s + (0.984 + 0.173i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2090, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.056721991 - 1.086983469i\)
\(L(\frac12)\) \(\approx\) \(2.056721991 - 1.086983469i\)
\(L(1)\) \(\approx\) \(1.477712692 + 0.001927544544i\)
\(L(1)\) \(\approx\) \(1.477712692 + 0.001927544544i\)

Euler product

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

−18.827951893292287632202005717765, −18.08464451275088421491874734013, −17.09313832716605819962173261477, −16.64091375923517967516964781221, −15.58729790017638765173489836540, −14.95501109961072055691991065959, −14.56980154259689194958672836374, −13.84823348291501716134271744188, −12.88661471638009596239041915497, −12.13742227317788973856630939484, −11.97261550231737696221602871456, −11.23313186407627691634148874467, −10.63768107571501145694484054481, −9.97110849003178645112905467802, −8.76363885336201099383832392325, −7.72662589315365769041182562809, −7.20936299653499574440897342547, −6.52652184993851999814925067090, −5.89599037053085810081798066126, −5.04118996361618033901103025356, −4.39842134888569479900620326594, −3.693689117233398600018138968832, −2.51619108029002650730397561327, −1.940046842449400920928066027771, −1.2379482761751915868938439808, 0.55221863912994049660314687210, 1.30848150850309462779886191425, 2.87503798978155337548277831604, 3.57417728235055472951632662675, 4.29048649020386191935366001576, 4.90221444581911416810890597516, 5.381105738568451813006827771151, 6.1504601654571204565732135178, 7.08688908148911171796206765203, 7.7736605197403381334999184708, 8.45779995009358393321566863297, 9.27445785043180624850453258368, 10.44052299743011663129949415298, 11.0137080570084698945719598504, 11.577571413736126754987544084122, 12.08084313452346566870902378763, 12.970789738248525820871472511869, 13.46571809618130488071479953692, 14.44033827419450858378192667820, 15.11950045482736817811251222522, 15.51843551301354670060198925276, 16.40556334296979246824664726194, 16.919946983355014258710370417526, 17.14891312717301169278629353986, 18.106244509328474838722243311183

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