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Properties

Label 1-4033-4033.3864-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.853 - 0.521i$
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 + (−0.835 + 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.597 − 0.802i)5-s + (−0.0581 − 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (0.893 + 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.835 + 0.549i)3-s + (−0.5 − 0.866i)4-s + (0.597 − 0.802i)5-s + (−0.0581 − 0.998i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (0.893 + 0.448i)12-s + (0.597 − 0.802i)13-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3864, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.853 - 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01259995884 + 0.04477662090i\)
\(L(\frac12)\) \(\approx\) \(0.01259995884 + 0.04477662090i\)
\(L(1)\) \(\approx\) \(0.5786192714 + 0.1990404725i\)
\(L(1)\) \(\approx\) \(0.5786192714 + 0.1990404725i\)

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 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (0.396 - 0.918i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (0.893 - 0.448i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.993 + 0.116i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (0.396 + 0.918i)T \)
97 \( 1 + (-0.686 - 0.727i)T \)
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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

−18.15959779472196597434803279913, −17.30758463738619847374514929446, −17.11683935867361971092535293844, −16.5982396149945243683348514884, −15.276269191370437479509366167704, −14.501632261874254553446424186295, −13.69837093468502484109816216401, −13.10626450081227727035756700064, −12.650721644789643531735318872107, −11.54474388490898892258802299044, −11.19207807990114146548419101642, −10.549213682363418552532628774741, −10.13201513167032945580409667610, −9.16542951693401769695938769621, −8.402314173297551134759544653517, −7.33597297962055603147095437174, −6.9705817422046046914283088069, −6.34475082275223829153277219033, −5.23408786359962093821275242772, −4.24770020587283835577129310268, −3.9032821434234046443683809811, −2.52898789152769550718959969207, −1.7616476028146240306184727358, −1.445745779134054982759894948654, −0.01864869848215504654708069771, 1.08045392435445840280234665102, 1.76366874985052437450050929300, 3.18503750726244381101969381759, 4.28011146453722174446773517848, 5.06028327446931033457080931577, 5.534228960655004006708329117062, 6.03803254681494183414048120699, 6.69524131712801136153443978417, 7.839663552110418935702935237646, 8.664527891022810234961841712283, 9.07768138383064676171103321038, 9.6058347702052813211922398993, 10.624927213960030942468663821677, 11.10394757213757789214123544517, 11.885816733981274961991348649178, 12.88987114619924537883995778992, 13.33804292316791145040615123717, 14.37713947124739948929509295276, 14.97399489569497636193558480934, 15.7878948321456238226879857234, 16.110845553227421309056963685362, 16.891510644564017971692057864386, 17.361035115803580407973686086074, 18.02222349978379382522213684509, 18.49761903855699212058988997422

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