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Properties

Label 1-1140-1140.599-r1-0-0
Degree $1$
Conductor $1140$
Sign $0.363 - 0.931i$
Analytic cond. $122.510$
Root an. cond. $122.510$
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  + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (0.939 + 0.342i)47-s + (−0.5 + 0.866i)49-s + (−0.766 + 0.642i)53-s + (0.939 − 0.342i)59-s + (0.766 − 0.642i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (0.939 + 0.342i)47-s + (−0.5 + 0.866i)49-s + (−0.766 + 0.642i)53-s + (0.939 − 0.342i)59-s + (0.766 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(122.510\)
Root analytic conductor: \(122.510\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1140,\ (1:\ ),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8402820375 - 0.5739820420i\)
\(L(\frac12)\) \(\approx\) \(0.8402820375 - 0.5739820420i\)
\(L(1)\) \(\approx\) \(0.8583272257 + 0.01768707689i\)
\(L(1)\) \(\approx\) \(0.8583272257 + 0.01768707689i\)

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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−21.37331419463578907516603404633, −20.4076727196538697357030027753, −19.80296338911406329852765181568, −18.81197560998546030761442096141, −18.28224947003230193958934813512, −17.58312866689829824494993839590, −16.33616821139962350788951562961, −15.95811960123206906017090819103, −15.13900884926524331235586948771, −14.311822317565475968110848464666, −13.202768005283142413010436660986, −12.843503613693320524421592471038, −11.81808902211837327068525553753, −10.98820235917296199989099361946, −10.24068668555999180515381377779, −9.20289505341169362999767273121, −8.55851564376064363640738622398, −7.7379420001583856069274915536, −6.60143494259319713002160757232, −5.80268976205826768334098639174, −5.162157735390694887768467717, −3.882638018417777781837177853215, −2.931405594845921636178112454707, −2.221757250234288528593832348556, −0.69751742581149190841329138780, 0.28434616347186033460452418077, 1.67278218038073387422057667233, 2.51758128250101682515994605786, 3.99301744689386585779584760363, 4.234404042220879696233934128624, 5.572702725915057840858887168533, 6.511019084908998790214576943619, 7.28124368225959761853011697774, 7.976576257820835723648150768504, 9.28573299831402487296158410267, 9.70537562028991664806419329163, 10.76137721154336953230708706918, 11.367833111444973690932887810950, 12.48379062522036344729228798468, 13.17363422547685499981776579639, 13.84500144803452605397864999043, 14.740928375668673143250306734593, 15.64283767053543417539185271899, 16.28752472080457266482893097190, 17.18218993672301734382286617039, 17.75796642104736169454587173257, 18.765520320293845405735254768, 19.438467690347513631054294200652, 20.34799235207135747102436183622, 20.71299145243789828257274339286

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