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Properties

Label 1-2268-2268.83-r1-0-0
Degree $1$
Conductor $2268$
Sign $-0.902 - 0.431i$
Analytic cond. $243.730$
Root an. cond. $243.730$
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  + (0.893 − 0.448i)5-s + (−0.0581 − 0.998i)11-s + (−0.597 − 0.802i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 − 0.957i)23-s + (0.597 − 0.802i)25-s + (−0.396 − 0.918i)29-s + (−0.686 + 0.727i)31-s + (0.173 − 0.984i)37-s + (−0.993 + 0.116i)41-s + (0.835 − 0.549i)43-s + (0.686 + 0.727i)47-s + (0.5 − 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)5-s + (−0.0581 − 0.998i)11-s + (−0.597 − 0.802i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 − 0.957i)23-s + (0.597 − 0.802i)25-s + (−0.396 − 0.918i)29-s + (−0.686 + 0.727i)31-s + (0.173 − 0.984i)37-s + (−0.993 + 0.116i)41-s + (0.835 − 0.549i)43-s + (0.686 + 0.727i)47-s + (0.5 − 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(243.730\)
Root analytic conductor: \(243.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (1:\ ),\ -0.902 - 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4343727437 - 1.915336977i\)
\(L(\frac12)\) \(\approx\) \(0.4343727437 - 1.915336977i\)
\(L(1)\) \(\approx\) \(1.122191594 - 0.4045075732i\)
\(L(1)\) \(\approx\) \(1.122191594 - 0.4045075732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.0581 - 0.998i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.286 - 0.957i)T \)
29 \( 1 + (-0.396 - 0.918i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.993 + 0.116i)T \)
43 \( 1 + (0.835 - 0.549i)T \)
47 \( 1 + (0.686 + 0.727i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.0581 - 0.998i)T \)
61 \( 1 + (-0.973 + 0.230i)T \)
67 \( 1 + (-0.396 + 0.918i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.993 + 0.116i)T \)
83 \( 1 + (0.993 + 0.116i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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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

−19.90723336606321209837258282696, −18.87795817098360479516945765683, −18.324170144722344211453177106382, −17.779019015730862393297293358, −16.81401133361856686211625398018, −16.500280274339264041507596098285, −15.24060774490380310928169445208, −14.785297815337792897643397762331, −13.90327246417196753322805954549, −13.54674731569572004471830598604, −12.37246702693138880447547757335, −11.94445338990141227942642603781, −10.96803725933493430186175750497, −10.05640818956710082320031083865, −9.61062638730698415023596240937, −9.02760465154753288906073673729, −7.54314104576574110573878157491, −7.33916086333296385763923844833, −6.32867174259131936769110321783, −5.47945441013515230867437022228, −4.867540951299649476971688849476, −3.75335662748240672115787999464, −2.861843098722196214811200442254, −1.95279167067648582726807628418, −1.30491534770324119951865043817, 0.32848447868989439263385657526, 1.081529959875597016684521304403, 2.19642921766331077143891943168, 2.97164894688317903927282381380, 3.91794500541956524299866986863, 5.06333376303681843773428687612, 5.5976914856102223986249094410, 6.244713293462861083429651999481, 7.309565876766924985145371808102, 8.14449672491856995835873551904, 8.85487146925950594228581188914, 9.62480926894186975586834688803, 10.35668963233138785446713897378, 10.976713051284728417438924389405, 12.0685647706533550342106568596, 12.6789846198071624986489042578, 13.38488403741775391613154705349, 14.0973927364872476330447737775, 14.699009358986209212095638567673, 15.69261278687003612393048090267, 16.42395084121870190066875978624, 17.02787180754481147560283466987, 17.70606645322774055748035445715, 18.380079412958556117996322810807, 19.199724659999427003347455524426

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