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Properties

Label 1-4033-4033.177-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.941 - 0.336i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.941 - 0.336i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.977801365 - 0.3429942894i\)
\(L(\frac12)\) \(\approx\) \(1.977801365 - 0.3429942894i\)
\(L(1)\) \(\approx\) \(1.245395383 - 0.7135373869i\)
\(L(1)\) \(\approx\) \(1.245395383 - 0.7135373869i\)

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

−18.35735216489876096948569320458, −17.391256928726761770647361014857, −17.063617418478854399580728878251, −16.29685582065867390402192739730, −15.81282606860358138442710341469, −15.27099804032444427124933853021, −14.31255803326572801291000889790, −13.71414605277716632048206757194, −13.3550406441173448917531270076, −12.848836139905580618349714394407, −11.59445876669894646796628338106, −10.893362602572178224240214845088, −9.75939490289801556854932903325, −9.58377231105280172614643174686, −8.8015651748608780992954272635, −8.089376530259474636638367103989, −7.11471006872532105261525692377, −6.67609179050624741097587024344, −5.58993415710831031603404133725, −4.92238979337105350982837250514, −4.51763894784284050631132638291, −3.627886258735362188398274985887, −2.83724527819538812652267581623, −2.03869410560328933978889323555, −0.42655922932595041149556604588, 1.12348474383853811459146410204, 1.759601472240225873649988978472, 2.80038348836922129708258908138, 2.907561307333498790329154149714, 3.72684198768340034910089030566, 5.22138388200541654929060471002, 5.71893295232122055570303994809, 6.238038002941729842462000158247, 7.02712376844106257573370585581, 8.29214744229163262535212590016, 8.65110317319791286222929627584, 9.58071974048115128126290173412, 10.17663678617384765811040740281, 10.9293165231319415699970573743, 11.61972178218820075640931833149, 12.58666223031787541969231308959, 13.01502467808801046903556383658, 13.37766025574775506184210357295, 14.122068933122368878209905688989, 15.02516856951941923063675462112, 15.17833050165949668169146412434, 16.33889896787519849429645771572, 17.536382532861501245248370655, 18.05377830820369124493352453605, 18.62019169657915893706258410932

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