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Properties

Label 1-4033-4033.3011-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.542 - 0.840i$
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 − 3-s + (−0.5 + 0.866i)4-s − i·5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s − i·5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.542 - 0.840i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.542 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2064648100 - 0.3788431460i\)
\(L(\frac12)\) \(\approx\) \(0.2064648100 - 0.3788431460i\)
\(L(1)\) \(\approx\) \(0.8087624042 + 0.1533308047i\)
\(L(1)\) \(\approx\) \(0.8087624042 + 0.1533308047i\)

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 \( 1 - T \)
5 \( 1 - iT \)
7 \( 1 + T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 - T \)
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 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 - T \)
61 \( 1 - iT \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.81038199925044117327608759361, −18.05217436912259611691823262411, −17.50822609573510412879071712662, −17.02511061409190234731269877120, −15.66055220776645275995422944236, −15.165130669339822028548053654433, −14.66214248800735804231276855945, −13.86913614436529658028030024514, −13.05972641789823181910943392691, −12.379601868412904834446061399492, −11.79320235514312063635894252634, −11.02649831484433397769417296128, −10.74614208335211410299591552037, −10.020132421621880268523740136564, −9.46102756389301382788387268544, −8.08075453327610379875726807749, −7.43582597427165902297820110185, −6.62712059325081785214765433730, −5.72156433552670777768835253775, −5.24818261208963309561888782280, −4.4479902874451051078368250869, −3.85027950141714895007468373844, −2.65509999449634970227501591905, −2.08797209763841375960950606311, −1.24327126722078923152895815862, 0.13227888951156300984528307535, 1.04757967559122730143037446670, 2.29065886248929441245040020438, 3.33003067695517496535880240276, 4.712411072181174603408662396371, 4.82760649460136085069064898960, 5.18857499278913818371463632866, 6.11211401663289103048180151277, 7.018134705635114411120210262467, 7.640310665019041003011673074743, 8.2265891892991057437992133730, 9.1922391023244790541754625282, 9.697356720738901980080479232495, 11.07632481339807997006522018600, 11.35136931137669900282816860528, 12.25083015476031664362368665490, 12.780030326157834298078806197718, 13.42780831172890742752875018140, 14.06762383351283272223933339376, 15.09171995427400336586912583756, 15.58912964527038894005720878708, 16.28196761033547799982802422809, 16.79203964713580650340991405752, 17.44944248929455450358481274857, 17.91738649568932274794403237167

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