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Properties

Label 1-1140-1140.143-r0-0-0
Degree $1$
Conductor $1140$
Sign $0.983 - 0.180i$
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 + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (0.642 + 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + i·37-s + (0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.342 + 0.939i)47-s + (0.5 − 0.866i)49-s + (0.642 + 0.766i)53-s + (−0.939 + 0.342i)59-s + (0.766 − 0.642i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (0.642 + 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + i·37-s + (0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.342 + 0.939i)47-s + (0.5 − 0.866i)49-s + (0.642 + 0.766i)53-s + (−0.939 + 0.342i)59-s + (0.766 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(5.29413\)
Root analytic conductor: \(5.29413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1140,\ (0:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.736852357 - 0.1579143613i\)
\(L(\frac12)\) \(\approx\) \(1.736852357 - 0.1579143613i\)
\(L(1)\) \(\approx\) \(1.221940208 - 0.04897636699i\)
\(L(1)\) \(\approx\) \(1.221940208 - 0.04897636699i\)

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

−21.338437963138415534759576469933, −20.80601453190606261017760764325, −19.71934018174843371192437259034, −18.989273690371394539632227116679, −18.163392960262315177583900295834, −17.70733677272090884997289544379, −16.56824980959826263271039798523, −15.973044081622730044141359285778, −15.06385944295180864800906258186, −14.39030220057893807379934181006, −13.51426261329626531290202436857, −12.7706017982225938840089242190, −11.82459987256761762045249818572, −10.92486864964701223406820383647, −10.60340169267789279419538270066, −9.17912582920414525297823830914, −8.46259716491186583962813779424, −7.99913136107175587791373684049, −6.68228539372809402454195028622, −5.9162597227193106520078162372, −5.07398348136659748635566999503, −4.121594268893337734415772483446, −3.08565506572216543490322047175, −2.06450883698451414507143747914, −1.00822928616672850758041875358, 0.94037741793112301360293706079, 1.95279996799602378797185508479, 3.03961962660153830898800089250, 4.18179717423930744789706879088, 4.89294682419268742278130256259, 5.76748360966170794123690586800, 6.99379211144384291799427173136, 7.55716906115405629784080944891, 8.47076787728073991230037148850, 9.33681110957233170389151349928, 10.321937913350877531212585705941, 11.027885814886008458965560169567, 11.72878696342103297003233795265, 12.73628815177355347050328108770, 13.5818464240269723055650536733, 14.14372017737165068728652334201, 15.23553671635496322598502977674, 15.68596734982681213284278159466, 16.73399372027003303412640598357, 17.58241267928971237715849858201, 18.08132454795897282047003152137, 18.86407734296248851069220862430, 19.961686383618570406140153693014, 20.66174822014262740233334659162, 20.9806793851510184626989993789

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