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Properties

Label 1-1140-1140.983-r0-0-0
Degree $1$
Conductor $1140$
Sign $0.966 - 0.255i$
Analytic cond. $5.29413$
Root an. cond. $5.29413$
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  + (−0.866 − 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.342 + 0.939i)13-s + (0.642 + 0.766i)17-s + (0.984 + 0.173i)23-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (0.984 + 0.173i)53-s + (0.766 − 0.642i)59-s + (0.173 − 0.984i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.342 + 0.939i)13-s + (0.642 + 0.766i)17-s + (0.984 + 0.173i)23-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + i·37-s + (−0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (0.984 + 0.173i)53-s + (0.766 − 0.642i)59-s + (0.173 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.966 - 0.255i$
Analytic conductor: \(5.29413\)
Root analytic conductor: \(5.29413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (983, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1140,\ (0:\ ),\ 0.966 - 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266417195 - 0.1646886018i\)
\(L(\frac12)\) \(\approx\) \(1.266417195 - 0.1646886018i\)
\(L(1)\) \(\approx\) \(0.9871672750 - 0.05291452312i\)
\(L(1)\) \(\approx\) \(0.9871672750 - 0.05291452312i\)

Euler product

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

−21.239267849160340713547598805372, −20.550529222496413724576329792696, −19.90711224938612405524335969745, −18.88675941824703641678537732232, −18.3812356519889372019457428379, −17.57020750084828164475353845024, −16.59894180398360544833613641587, −15.89380192591586920684408027056, −15.181535395200337616971222028038, −14.50133456974125764310647396900, −13.21327809164237120456803212188, −12.860934469079188884966619106386, −12.05510269441550252042229587519, −11.032229599940489109476390031106, −10.168033017285951013406867642429, −9.47033523369873446204500907084, −8.680778422467360197201677684191, −7.54225742990418681552403991521, −6.98900538783210481511466030435, −5.73042861927033614325147184927, −5.28805427911667676805888050298, −4.00771267808858509353262252819, −3.02554285535688843632157699876, −2.3096926106617890499530290667, −0.82113611055932461602195615445, 0.75339325721032233249817728744, 1.99510895680207284931602473409, 3.30023035765049371131095510843, 3.75519979416617709753009006055, 5.01419056070211238435218422859, 5.96556588175887880147634600161, 6.71205782774211540466636859941, 7.576136366705366096290036107876, 8.5862678879472197599482420567, 9.31255097929441436556113810844, 10.28265762733458235296832860557, 10.90814047321675681446611129523, 11.83558734010148978996947694098, 12.79359171410105594109564422763, 13.47411835598635340838819938419, 14.09926661436841234095001893061, 15.1538050191516675879190827924, 15.93955081746326327547599071043, 16.74496141995958359362939234299, 17.10456092270140924311330534484, 18.495349916333545241211679769734, 18.95361827426320401023943612192, 19.59201191031130754204341065812, 20.59578347940446517902707816572, 21.29103503476933357975470136706

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