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Properties

Label 1-1143-1143.617-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: $\chi_{1143} (617, \cdot )$
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

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

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