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Properties

Label 1-4033-4033.3480-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.885 - 0.464i$
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 + (0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.984 − 0.173i)5-s + (0.5 − 0.866i)6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s − i·10-s − i·11-s + (−0.766 − 0.642i)12-s + (0.766 − 0.642i)13-s + 14-s + (0.984 + 0.173i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.939 + 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.984 − 0.173i)5-s + (0.5 − 0.866i)6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s − i·10-s − i·11-s + (−0.766 − 0.642i)12-s + (0.766 − 0.642i)13-s + 14-s + (0.984 + 0.173i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.885 - 0.464i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3480, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.885 - 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.466351493 - 0.8532712512i\)
\(L(\frac12)\) \(\approx\) \(3.466351493 - 0.8532712512i\)
\(L(1)\) \(\approx\) \(1.826286691 - 0.5378680669i\)
\(L(1)\) \(\approx\) \(1.826286691 - 0.5378680669i\)

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

−18.30842727588325117941492060482, −17.915080087097249573058376158840, −16.950292893932521268280077285421, −16.733947836147178104262179973160, −15.65752979921891324970175897223, −14.81736190955310896995336402074, −14.5449834927933233185868279931, −13.817931342751489752169786251850, −13.20793306664098663616282399202, −12.95144005026625733881370985816, −11.89315541784941829817631826401, −10.64634257206152309705400592301, −9.99298730083922641035748561554, −9.28397241984991269477701624516, −8.83583571011465444879127754510, −7.85424050989127538122303349295, −7.225848529054717939393304315188, −6.79761564021947833840259001835, −6.03064825543173864298919044837, −5.07621321677954790620815432819, −4.23529267824757112076639404028, −3.68306569184716310241603320723, −2.66441580308953648458335843404, −1.724928318481222419777998492261, −0.905458226170877431609606785175, 1.21066898804131920879141796848, 1.65349772801577930126429473209, 2.63953620740654016858373237077, 3.217799368055691763309602796743, 3.73936924573019638235478071259, 4.97434156991009523099436428645, 5.62034455388763126182904889152, 5.92325737402236637124895965697, 7.531336792110689380517787768444, 8.326532240337121648625208297173, 8.95239904947260081615627126965, 9.34345517985665465061300704534, 10.09269400013908935221467092895, 10.784223864867612572268802186972, 11.40960209965970471640714309314, 12.57276513302622058495519448316, 12.8151739958863944323552499928, 13.73110499581547863289229009620, 14.161200315156588441752235882876, 14.70065153185665829312324800744, 15.580211418886730746357196708431, 16.324557270616295168605245963903, 17.15328631641784404826974051990, 18.17909777552995841297058736319, 18.55184244707617117207792073834

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