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Properties

Label 1-1080-1080.317-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.475 - 0.879i$
Analytic cond. $5.01549$
Root an. cond. $5.01549$
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  + (−0.984 + 0.173i)7-s + (−0.939 − 0.342i)11-s + (−0.642 + 0.766i)13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.984 + 0.173i)23-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (0.984 − 0.173i)47-s + (0.939 − 0.342i)49-s − i·53-s + (0.939 − 0.342i)59-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)7-s + (−0.939 − 0.342i)11-s + (−0.642 + 0.766i)13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.984 + 0.173i)23-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (0.342 − 0.939i)43-s + (0.984 − 0.173i)47-s + (0.939 − 0.342i)49-s − i·53-s + (0.939 − 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.475 - 0.879i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.475 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7787021655 - 0.4643457320i\)
\(L(\frac12)\) \(\approx\) \(0.7787021655 - 0.4643457320i\)
\(L(1)\) \(\approx\) \(0.8452748917 - 0.06277503893i\)
\(L(1)\) \(\approx\) \(0.8452748917 - 0.06277503893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
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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

−21.53671872999737676609963418412, −20.76091157006525158559201979181, −20.09521955756870532716837856212, −19.16921182650491892490039628515, −18.6328907456183244272080613847, −17.71066745065646830802416038685, −16.80144521391715837502440864371, −16.2246192255814978209701472253, −15.30890983952809448769572296679, −14.66709217772574432381347891188, −13.612249171456306830755956750790, −12.750936198615653984543236258170, −12.4349570883962030398330789250, −11.20450006242186787101991042624, −10.15041340006422848962368986976, −9.90883240756439827245258922644, −8.765542071935542785469327121050, −7.69756034389699268392936177248, −7.17484196060393750872056071307, −5.99646840841196417172030921319, −5.31038556700691706414161191463, −4.2427731743984194500668970680, −3.109210632254407118148213433584, −2.52627901169399279947652330022, −0.96859820049275085249609013474, 0.44974090997619240681016587559, 2.06039707248223525737509783528, 2.92688049931352413231655863285, 3.81892528366099775913767740909, 4.986736824042968828802881267039, 5.76620055466582627619926479512, 6.75808475461726525834805665894, 7.45632462433344759970834627555, 8.54677190183741654641304263562, 9.35117266133757908197106248177, 10.09591965483667628423186897341, 10.951940728236565940842191231174, 11.85635410536304602250634766410, 12.84357283658524748674805530019, 13.21790312038100627540362276338, 14.280506001349889792094098519620, 15.18986545485410851030166908192, 15.793888861166287670824383023124, 16.82769574711096085655421486865, 17.12020785228054778857633297287, 18.622181400509372828801383042730, 18.83555775978284609651709899492, 19.66158160935921912555806085485, 20.57579491038833034493449907057, 21.44718466524981358219844921285

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