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Properties

Label 1-4033-4033.1014-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.962 - 0.269i$
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 + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.342 + 0.939i)5-s + 6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.342 − 0.939i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.342 + 0.939i)5-s + 6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.342 − 0.939i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03088108016 - 0.2245380048i\)
\(L(\frac12)\) \(\approx\) \(0.03088108016 - 0.2245380048i\)
\(L(1)\) \(\approx\) \(0.4084999268 - 0.06875739183i\)
\(L(1)\) \(\approx\) \(0.4084999268 - 0.06875739183i\)

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

−18.89338711571579302119546187365, −17.80308046419618017178579024073, −17.48840703726348987814657941179, −16.77158181428275934477702455780, −16.09907636647180598957168235862, −15.77497786240939520061787465328, −15.03602565967194816217933546037, −13.873678571860698027211694205561, −13.26159412027172955284541065704, −12.51505849353450860065856808427, −12.0487225759647948747408568648, −11.176450449565800152020074788055, −10.36168216824204711495170594405, −9.77315225036268200074810703853, −8.87072151829279078148757059851, −8.3786580662848778352893221474, −7.46971287143380567266362908309, −6.99501136472249435046545024728, −6.10362971434301335591853539581, −5.659083290886143918228816260460, −4.79619141572789975000028334093, −4.13348588780006277268035600705, −2.520471740656203626020244724184, −1.824195678606412378968989498092, −0.90827543425281711730585917901, 0.16408285765110005425770726583, 0.736629317705070961445212648909, 2.469544015626680934392918948913, 2.93362806228829166274216991869, 3.68691466310646270380844564050, 4.339069517087141221200923956424, 5.49530406248201776601865306262, 6.322329779083482198405363601439, 6.92634827364255185174398628458, 7.76831901154077303661935743572, 8.41530918723123894542118142115, 9.53806523679103891608276456387, 10.07623446238491518602409433554, 10.51636904010455791683038242493, 11.01305239054773980050211290573, 11.77375013079175750787127047407, 12.5409938863226570965346271572, 13.02930831500079781196636928921, 14.06613194607277173009518122849, 14.97729376742745802702278334022, 15.848859741241963900033816850411, 16.124169936217280089280623598272, 16.76436252344918081812339627751, 17.691105287197647595907226245667, 18.13475383379553895008948262073

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