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Properties

Label 1-4033-4033.2826-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.177 - 0.984i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s − 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s − 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2826, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.177 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1553991878 - 0.1299000883i\)
\(L(\frac12)\) \(\approx\) \(0.1553991878 - 0.1299000883i\)
\(L(1)\) \(\approx\) \(0.5511122720 + 0.03535148194i\)
\(L(1)\) \(\approx\) \(0.5511122720 + 0.03535148194i\)

Euler product

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

−18.63588174107970115279675476622, −18.19847585927434510931004277930, −17.18087174373084201591487208761, −16.53911542807525720919463079303, −15.80939918098758846970445325139, −15.42002648138709113981715805773, −14.6558007902887726884711133401, −13.872772001819605935564665343869, −13.29326916254145393218022210869, −12.52293202877929496810979504421, −11.86258429229564254959704276236, −11.213141154774732107149658428210, −9.98327110667605782894869667729, −9.28819300051260449973101786988, −8.70997301844496744294638167065, −8.1515454970034398611586249151, −7.521469427886582696251885215495, −6.85379495081485068512375406675, −6.143608994946637927842604252967, −5.32312861358597344655390422916, −4.61469011331438757478226768097, −3.53239755594191091051438632499, −2.59215032585677511057779403523, −1.816326551962784374757684662212, −0.56215438015905713385255221205, 0.10843922836699329462215720430, 1.59034294704270059219022214026, 2.60352735710574262038860714513, 3.29818094762110653591548989672, 3.743066925107360597206415918372, 4.4566725130570144378057808439, 5.22904613134413313812019547792, 6.53144190305453590633330395094, 7.71123113977930025406317617159, 7.82143356503535804799260722061, 8.52520136739692910470808724288, 9.63717997603762052793446697141, 10.30502109962856070186312883914, 10.43028927486013994414763209028, 11.08231520813527074027178568074, 12.246388644325230963404936641084, 12.59764129902518862377651165922, 13.47230278343787253213646077126, 14.31647002619196914063518624677, 14.87898695778645787195241615422, 15.71820266459423198299464592367, 16.33681080907161547746329893940, 16.86647163630297556447245101215, 17.784625486575718151539056258568, 18.41806899472080021747127552753

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