↑

Properties

Label 1-6048-6048.1699-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.527 + 0.849i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (−0.422 + 0.906i)5-s + (0.906 − 0.422i)11-s + (−0.906 − 0.422i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (−0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 + 0.939i)41-s + (−0.819 − 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 + 0.906i)5-s + (0.906 − 0.422i)11-s + (−0.906 − 0.422i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (−0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (0.342 + 0.939i)41-s + (−0.819 − 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.527 + 0.849i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9952073100 + 0.5534067225i\)
\(L(\frac12)\) \(\approx\) \(0.9952073100 + 0.5534067225i\)
\(L(1)\) \(\approx\) \(0.8938784708 + 0.1016276837i\)
\(L(1)\) \(\approx\) \(0.8938784708 + 0.1016276837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.422 + 0.906i)T \)
11 \( 1 + (0.906 - 0.422i)T \)
13 \( 1 + (-0.906 - 0.422i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.422 - 0.906i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (-0.819 - 0.573i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.996 + 0.0871i)T \)
61 \( 1 + (0.422 + 0.906i)T \)
67 \( 1 + (-0.819 + 0.573i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.422 + 0.906i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
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

−17.474416005408879327722654879237, −16.91785101137036102852537911386, −16.34117051535487400298877021698, −15.7400326879153884962703428900, −14.9351179819713921787028704435, −14.39303144927035444971841762294, −13.69622241600029481240734169430, −12.804194061181527291748400556650, −12.317338391521602823422462756251, −11.82463731289670057637897934108, −11.131975661578104217121638221533, −10.20064777208893493927121286443, −9.463442421206575687273123561256, −9.01820706085779553245805819168, −8.30148829695068390175801023767, −7.43002685026127602770572973108, −7.04189074212047719156044507272, −5.97649345776146693886010228610, −5.345097099332783383255683479569, −4.55092387047137321426912003020, −3.94931981714189280472621073932, −3.34236076059468408619156548957, −1.85964830385434021231347310346, −1.73126740106635219315754425260, −0.400040403085365509761008991574, 0.6751731260280373707726159684, 1.83974991672478691024554720350, 2.71146087714584453531712298304, 3.26369761815010357878857285759, 4.07002236032978060681590599380, 4.78641981920791433188961082407, 5.73845672319580611149749953334, 6.3496259241219574978938652939, 7.29992366769825268731049835653, 7.432191145290434558825633440622, 8.44158518740462354209520351137, 9.242431819472707244716868975732, 9.86336586858845293948962698172, 10.531599857123589101357342603023, 11.33427141063027346778727920463, 11.827107194872836378676137064846, 12.28613161547662038295674598383, 13.456463610909783537537977764722, 13.92101703163655394920366413388, 14.529572687424684652379343572549, 15.20759568629153966056619677410, 15.734784890751979408829223515, 16.51079008952211231506008867028, 17.14126378215383157232409602829, 17.99625582458409808964012622609

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