↑

Properties

Label 1-2736-2736.1915-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.860 + 0.509i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)5-s + (−0.5 + 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + 31-s + (−0.342 − 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)5-s + (−0.5 + 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + 31-s + (−0.342 − 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1915, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.313914439 + 0.3594609497i\)
\(L(\frac12)\) \(\approx\) \(1.313914439 + 0.3594609497i\)
\(L(1)\) \(\approx\) \(0.9474860678 + 0.1714864111i\)
\(L(1)\) \(\approx\) \(0.9474860678 + 0.1714864111i\)

Euler product

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

−19.47820980329193711998775719771, −18.39891188596047442066910962300, −17.83622692085234383362955495388, −16.8571968900596224146770893110, −16.52545694052821516121724285296, −15.572897161175524844320699838783, −15.28609385433155160163350394924, −14.05355710538236436109914216645, −13.5414841838592058030985397039, −12.682068870926648971091215526395, −12.197150739702476605050124219319, −11.45684421391548979377312543197, −10.3783991972753396446494784299, −9.98544557395868857578638348350, −9.0848021888642362648441230866, −8.21052804534563523041518147988, −7.576200843456634457984811101755, −6.9578557299935545125460568542, −5.931884533989538826501270009826, −5.052979754799074881664848266494, −4.311027431935100277617774527315, −3.62759418025398127746371627130, −2.7968627119647074075445040771, −1.41105564551223161672968328849, −0.73484106966835465587600244783, 0.67644824431428679432247158796, 2.07233702882602612280710715070, 2.82747413156449461300837466732, 3.667165150661728673792459248189, 4.204998578035180781594121864279, 5.58798078542708293027434756325, 6.16399628834476402377911055122, 6.70236786961044231633047224963, 7.87934134167666322163268124138, 8.33042152061634499568046948951, 9.16731864492989891965442518163, 10.02569502809461092645301403184, 10.80286100614697874832939809448, 11.49479814051745495078610876284, 12.09861319256380870260294612130, 12.77544461994197539332539245667, 13.99018071823804563649114641103, 14.15142360724931036525167630998, 15.251200325864578206040576196998, 15.79216757972837172881011262859, 16.28547326161400837150092077608, 17.16951387378567215172322602236, 18.190964333229824876533124742422, 18.73724703501097331346462366799, 19.19994292452930640327719564859

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