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Properties

Label 1-2736-2736.85-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.851 - 0.523i$
Analytic cond. $12.7059$
Root an. cond. $12.7059$
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)5-s − 7-s + (0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)35-s + i·37-s + (0.939 − 0.342i)41-s + (−0.342 − 0.939i)43-s + (0.766 − 0.642i)47-s + 49-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)5-s − 7-s + (0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)35-s + i·37-s + (0.939 − 0.342i)41-s + (−0.342 − 0.939i)43-s + (0.766 − 0.642i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.851 - 0.523i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.851 - 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.648755343 - 0.4664637801i\)
\(L(\frac12)\) \(\approx\) \(1.648755343 - 0.4664637801i\)
\(L(1)\) \(\approx\) \(1.147948635 - 0.07506056235i\)
\(L(1)\) \(\approx\) \(1.147948635 - 0.07506056235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 - T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−19.37341685817830822140507618169, −18.63575534401770469000824160303, −17.81818585416509630694245647405, −17.17289736312740581569087940064, −16.56968284703099396567781170859, −16.01148050976014560752712934573, −14.84454386961450760827190309382, −14.47194447280575306388265165088, −13.54746506055191775601664752830, −12.91368053966321784781477625294, −12.42481447137375115492936458994, −11.41850635727513850030697798733, −10.690702134524367860575658601577, −9.65444614778696707508983808489, −9.335349691673089177675993347239, −8.86798242802338607095031304272, −7.43150611387980521540020318506, −6.75901064868992445083312671308, −6.29572390229125912566904462026, −5.33289639655914329762857024012, −4.51795597868477995048461123719, −3.70455720416282878352750083239, −2.56074081749804035399800951243, −2.03871278758211476005089434660, −0.89396516049344009837652770296, 0.64434804164611893850249061832, 1.773443394404083321738789106914, 2.68881200757885705659133893326, 3.35234814182311533722771009738, 4.29972103156335141039947872535, 5.43503580470242194416945144386, 5.92814501696043043550323879186, 6.778986928445127486074759467377, 7.25189310655127673475616158512, 8.61857309301998801306633651922, 9.111470429268413295245521046761, 9.832651001259776488876779749852, 10.47897247376499217863353304862, 11.21542760561247490082689205155, 12.21374835086600580805871143000, 12.9248485563016339072623065873, 13.47120246451650718407274120983, 14.12193621050218035873480435435, 15.05087189203454239701836763342, 15.528968426099549491900165215394, 16.6413235010149791590518587324, 17.0692367964647993261991028515, 17.64289203190187246526963873849, 18.560913220675241562824928296691, 19.200256040980421160654570664219

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