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Properties

Label 1-3744-3744.1213-r0-0-0
Degree $1$
Conductor $3744$
Sign $0.164 - 0.986i$
Analytic cond. $17.3870$
Root an. cond. $17.3870$
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  + (0.258 + 0.965i)5-s − i·7-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + i·23-s + (−0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (0.5 − 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.965 − 0.258i)37-s − i·41-s + (0.707 − 0.707i)43-s + (−0.5 − 0.866i)47-s − 49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)5-s − i·7-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + i·23-s + (−0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (0.5 − 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.965 − 0.258i)37-s − i·41-s + (0.707 − 0.707i)43-s + (−0.5 − 0.866i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(17.3870\)
Root analytic conductor: \(17.3870\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (0:\ ),\ 0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5915309401 - 0.5009031260i\)
\(L(\frac12)\) \(\approx\) \(0.5915309401 - 0.5009031260i\)
\(L(1)\) \(\approx\) \(0.9175369751 + 0.1090039261i\)
\(L(1)\) \(\approx\) \(0.9175369751 + 0.1090039261i\)

Euler product

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

−18.907948356426651966507238751370, −17.72219760303850033777605225948, −17.46330298877605248292642862876, −16.70958776834190364467540880451, −16.27045240627707781907362766504, −15.30693596440088897631629775839, −14.65900085007896865624686712191, −13.92710293580990274103497257626, −13.057216420090450968385408506360, −12.73229214231143809976012884448, −12.07258596062275084004361714153, −10.99964385313719703440434990520, −10.34045855164197216927075031767, −9.82075413528956608544804840875, −8.95678694455247923257993505596, −8.21392521849713425634025840953, −7.61169396189024359800499105583, −6.71500919987955567496197306668, −6.0377424166714869291570583030, −5.014259646164202740034059175491, −4.44764393836936979352231052374, −3.88717596540012867059831106027, −2.69803029213053008558954290618, −1.718263982495275808448720990940, −1.081883868190776994710746047762, 0.22092340278771093005087680477, 1.772455925642172418582565916155, 2.39880417261707163419749968588, 3.22990472926131534689610814432, 3.79009792336166537493481683752, 5.145554993984778964105942623838, 5.704383591460592318609508304585, 6.26220967033864825079206410221, 7.19449354399474063562133761050, 7.83951190409044468175400441809, 8.767976279518291934832194150, 9.31363961612446597949994781671, 10.19826390913958104750383472534, 10.821994918748034502203777100708, 11.58277176159791442712031582776, 12.04742829013452201949074940488, 13.09277481129557811352206860478, 13.69517934450731915803716921266, 14.36694555197281559725622246057, 15.10626406680296592957626414043, 15.56708527221312781690526434030, 16.38281655892890383169516149550, 17.13981128160316392460057314763, 17.95958095011402836106102215445, 18.508827801769988627877476913653

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