↑

Properties

Label 1-2664-2664.389-r0-0-0
Degree $1$
Conductor $2664$
Sign $-0.159 + 0.987i$
Analytic cond. $12.3715$
Root an. cond. $12.3715$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 − 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (−0.984 − 0.173i)19-s + (−0.866 + 0.5i)23-s + (−0.766 + 0.642i)25-s + (0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.984 + 0.173i)35-s + (0.173 − 0.984i)41-s + (−0.866 + 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (−0.984 − 0.173i)19-s + (−0.866 + 0.5i)23-s + (−0.766 + 0.642i)25-s + (0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.984 + 0.173i)35-s + (0.173 − 0.984i)41-s + (−0.866 + 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ -0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1063666151 + 0.1249257499i\)
\(L(\frac12)\) \(\approx\) \(0.1063666151 + 0.1249257499i\)
\(L(1)\) \(\approx\) \(0.7585431822 - 0.2192755646i\)
\(L(1)\) \(\approx\) \(0.7585431822 - 0.2192755646i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.13917728228059924570631486248, −18.357697613696450320863304503876, −17.879090918345086766436209947449, −17.123636898221716816380732021076, −16.20702025724596560686651936919, −15.32067389740730253582280051262, −14.97224824922215182166393697016, −14.34890956727427669322001799130, −13.55025074909861522672037253981, −12.38855647474892004232801831963, −11.96302738383135921113464532323, −11.43805178166847545809503388023, −10.35562553293041235653994541970, −9.8220604136507480544377047689, −9.02597008340267074067501889821, −8.09680971081403518680871358163, −7.45093647836336975994082365073, −6.50353884638863420583750250648, −6.159679568431212537349951508074, −4.74327599563276546091203948567, −4.45711278273509181058416587795, −3.166008590786255852951666880659, −2.42501865508862742996717799572, −1.878478285048580787266842078884, −0.054062523317448217638035289405, 1.0403569031286043170706319099, 1.83279853234796858894215279018, 3.10998516183033193670900258034, 4.04755177324360537193800351108, 4.48832919642972825567015984412, 5.38992162873383024579039165098, 6.340795110537366190108157446757, 7.07612542821534744330847290319, 8.12413115058605738957007764469, 8.39728440688342327969987905899, 9.34543116124594793184746351051, 10.2347796288342088834054946958, 10.81183550129270327352020847765, 11.77888383537995203025953940880, 12.29811365657586243566705157061, 13.1818461446410143256782275184, 13.70550427730226016461956690225, 14.54958617335519323414791757574, 15.247977896024993628719360612466, 16.215642003885223226981992171716, 16.65166048738450244918885511240, 17.38768613639008850646495903920, 17.75964643117056677742347141220, 19.19815213627936393017082560648, 19.62023575357459646130607137432

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