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Properties

Label 1-2240-2240.139-r0-0-0
Degree $1$
Conductor $2240$
Sign $0.290 + 0.956i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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  + (−0.923 + 0.382i)3-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s − 33-s + (−0.382 + 0.923i)37-s + (0.707 + 0.707i)39-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s − 33-s + (−0.382 + 0.923i)37-s + (0.707 + 0.707i)39-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7792040584 + 0.5778971156i\)
\(L(\frac12)\) \(\approx\) \(0.7792040584 + 0.5778971156i\)
\(L(1)\) \(\approx\) \(0.7884881749 + 0.1373614916i\)
\(L(1)\) \(\approx\) \(0.7884881749 + 0.1373614916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (-0.382 + 0.923i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.923 - 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 - T \)
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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

−19.47577589653331838653617175744, −18.79116693204958277387635295051, −18.084140776733968711912479133, −17.2982358166996030176625590144, −16.76130023262743882929686952113, −16.21697411466959088876471831004, −15.2837504543107253483185972752, −14.37369367734852859206317530521, −13.77743135018603916245192183334, −12.77296305354666289163628276443, −12.32565945235927820243903826339, −11.32535119766162088339833750, −11.126607507578795710976362960135, −10.03568442257211011597714766308, −9.26958631605783003668750103660, −8.45035917055089008472943714294, −7.385972549080334965103911722436, −6.81539039354293507555861184551, −6.05564841160335848381562748550, −5.38700265776271869154115165861, −4.31889692531649400006489061623, −3.800007045327863503683980927839, −2.29760321202779567180134790374, −1.61310820080629954430312443709, −0.46242729340598906614338576501, 0.90122504551037264589073816394, 1.84196300505897193386755656289, 3.19496912406289029543359625618, 3.9338256984684775657191409166, 4.79884349020787205682657044011, 5.60651767206631161318879755144, 6.13677565542242926042878918637, 7.27562952722939036580186575189, 7.64563527693331342028680141934, 9.107323355571110966471929556151, 9.542702949872807177568052885983, 10.36219961696879629829999115747, 11.044754014433542223032101375071, 11.98358183148857962229089328897, 12.23343130497687913775710821692, 13.19773668788421744895887928874, 14.17707835306811340938850966184, 14.8778343039153863922867449037, 15.61682221598122139412310130690, 16.34739716588161698204087174055, 16.92436496998030048092662195406, 17.72469006972604090276063008075, 18.15343149537584726552235676639, 19.024981787790430004765841800654, 20.04470944479007288759306034842

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