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Properties

Label 1-3744-3744.83-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.432 - 0.901i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.965 + 0.258i)5-s + (0.5 + 0.866i)7-s + (−0.258 + 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 − 0.5i)25-s + (0.258 − 0.965i)29-s + (−0.866 − 0.5i)31-s + (−0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s + (0.5 − 0.866i)41-s + (0.965 + 0.258i)43-s + (−0.866 + 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (0.5 + 0.866i)7-s + (−0.258 + 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 − 0.5i)25-s + (0.258 − 0.965i)29-s + (−0.866 − 0.5i)31-s + (−0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s + (0.5 − 0.866i)41-s + (0.965 + 0.258i)43-s + (−0.866 + 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.432 - 0.901i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.432 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2569717808 - 0.1618393205i\)
\(L(\frac12)\) \(\approx\) \(0.2569717808 - 0.1618393205i\)
\(L(1)\) \(\approx\) \(0.7550607664 + 0.2164043742i\)
\(L(1)\) \(\approx\) \(0.7550607664 + 0.2164043742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (-0.258 + 0.965i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.965 - 0.258i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.54560190299184204631023906508, −17.932197292189120300244595999331, −17.058032857407529589031973734774, −16.50524629029059804409455707830, −15.9278656951787564441534749422, −15.13517445932482814668127501619, −14.5176831946601225969617277529, −13.74806074134559994284624019495, −12.97399562889690518677113451846, −12.51945448765297210886824827838, −11.30884731464188140194917637147, −11.033446794739174791346557112030, −10.60271552786481138382746023424, −9.273701769290052228797522516902, −8.651824829835232773384790149411, −8.11858492293216079322324157485, −7.21495938352847023681655888680, −6.7894968318662959223402553771, −5.69532301301929529684857132382, −4.68936467989839805238037015744, −4.359491790930135671666864692690, −3.424416822940365781255704846066, −2.68559292307935927758409918576, −1.44867586732602302883971502281, −0.598881405680340913065074978151, 0.06973599796210082437307015321, 1.41453915571342333181126219114, 2.31560886256669693747167930049, 2.93279406949336168811296353559, 4.199113076330754791209380414139, 4.43795518523516370948288227696, 5.472737654260606263265940129, 6.25306756081628166487475183952, 7.18560687319321036201812577936, 7.713205091157354764160153460701, 8.48975683381161116370308169973, 9.09609251161799909843870686501, 9.982551723832364885676148744362, 10.86730376583082422892090136398, 11.41516526990120220232574242863, 12.04668923278441418143434911240, 12.73301509304720135074202505797, 13.34308934044795908290782965366, 14.66039310664249701232930063363, 14.82172110941611651514539501916, 15.53265410525001110996177447065, 16.041936019496208306269461422206, 17.050725247205715732447408413277, 17.73247046895254835225911649479, 18.32608878901542524892726836960

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