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Properties

Label 1-1143-1143.515-r0-0-0
Degree $1$
Conductor $1143$
Sign $0.782 + 0.623i$
Analytic cond. $5.30806$
Root an. cond. $5.30806$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.0747 + 0.997i)5-s + (−0.698 − 0.715i)7-s + (−0.623 − 0.781i)8-s + (0.733 + 0.680i)10-s + (0.124 + 0.992i)11-s + (0.698 + 0.715i)13-s + (−0.998 − 0.0498i)14-s + (−0.988 − 0.149i)16-s + (0.411 + 0.911i)17-s + (−0.5 − 0.866i)19-s + 20-s + (0.766 + 0.642i)22-s + (−0.124 + 0.992i)23-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.0747 + 0.997i)5-s + (−0.698 − 0.715i)7-s + (−0.623 − 0.781i)8-s + (0.733 + 0.680i)10-s + (0.124 + 0.992i)11-s + (0.698 + 0.715i)13-s + (−0.998 − 0.0498i)14-s + (−0.988 − 0.149i)16-s + (0.411 + 0.911i)17-s + (−0.5 − 0.866i)19-s + 20-s + (0.766 + 0.642i)22-s + (−0.124 + 0.992i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1143\)    =    \(3^{2} \cdot 127\)
Sign: $0.782 + 0.623i$
Analytic conductor: \(5.30806\)
Root analytic conductor: \(5.30806\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1143} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1143,\ (0:\ ),\ 0.782 + 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552464266 + 0.5428210229i\)
\(L(\frac12)\) \(\approx\) \(1.552464266 + 0.5428210229i\)
\(L(1)\) \(\approx\) \(1.326035633 - 0.1637621505i\)
\(L(1)\) \(\approx\) \(1.326035633 - 0.1637621505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (-0.698 - 0.715i)T \)
11 \( 1 + (0.124 + 0.992i)T \)
13 \( 1 + (0.698 + 0.715i)T \)
17 \( 1 + (0.411 + 0.911i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.124 + 0.992i)T \)
29 \( 1 + (-0.318 + 0.947i)T \)
31 \( 1 + (-0.411 + 0.911i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.583 - 0.811i)T \)
43 \( 1 + (-0.980 + 0.198i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.542 + 0.840i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + (-0.878 - 0.478i)T \)
71 \( 1 + (0.797 + 0.603i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.921 - 0.388i)T \)
83 \( 1 + (0.542 + 0.840i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (0.318 + 0.947i)T \)
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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

−21.0087400014989668534986505377, −20.94483061626184816347743063849, −19.8480119753798322085037783245, −18.77704035682518478577823369642, −18.12611747373822460958371937985, −16.907539502508580448874754550390, −16.49191205203008591532913999113, −15.86006459283021310021795057415, −15.14876060522815382663635552079, −14.11386410824490648913732393706, −13.36586608046285821376751050129, −12.75753234765250804663680686673, −12.0848775874296387839542923710, −11.29040996757952654704002174075, −9.96847401192693430268849678000, −8.93740705053040141674858882046, −8.43129777018212776349599662901, −7.63883115923171223023846584139, −6.25454691911670766667888082394, −5.87417503099332493329663822615, −5.1152177737233043460627005750, −3.99935902728771483670446712890, −3.254119850195555148861990001660, −2.192862670850754237934723504662, −0.51145188775984604130962529166, 1.404665919183796419865559386573, 2.23907170161359599514411934991, 3.48939784403979475013275246397, 3.77318179926124662668001108138, 4.92002729657039197522606223535, 6.060019267682295878113355293783, 6.78345608455906929590248878138, 7.31745973126550390654078741875, 8.95625695324460443047198955331, 9.76087756324267590376392002392, 10.580075579327800277994467092906, 10.95769551885100966309264412848, 12.04981586960810390560043817449, 12.76907455163664650958555193989, 13.677595983529957359044705267961, 14.12657950596822520154462159280, 15.12724794955641091747202164224, 15.58190403250198656059123255401, 16.748181747433888667696858911975, 17.71282073810159775306832254282, 18.50385170361518393769479438827, 19.375733622499556168984529309044, 19.74254262858061587261145477675, 20.664831041635259888814845209998, 21.598138259787594532245258927763

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