↑

Properties

Label 1-6008-6008.4813-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.892 + 0.450i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (−0.985 + 0.166i)3-s + (0.699 − 0.714i)5-s + (−0.637 − 0.770i)7-s + (0.944 − 0.328i)9-s + (−0.913 + 0.406i)11-s + (0.957 − 0.289i)13-s + (−0.570 + 0.821i)15-s + (0.146 + 0.989i)17-s + (−0.604 + 0.796i)19-s + (0.756 + 0.653i)21-s + (0.387 − 0.921i)23-s + (−0.0209 − 0.999i)25-s + (−0.876 + 0.481i)27-s + (0.756 + 0.653i)29-s + (0.944 − 0.328i)31-s + ⋯
L(s)  = 1  + (−0.985 + 0.166i)3-s + (0.699 − 0.714i)5-s + (−0.637 − 0.770i)7-s + (0.944 − 0.328i)9-s + (−0.913 + 0.406i)11-s + (0.957 − 0.289i)13-s + (−0.570 + 0.821i)15-s + (0.146 + 0.989i)17-s + (−0.604 + 0.796i)19-s + (0.756 + 0.653i)21-s + (0.387 − 0.921i)23-s + (−0.0209 − 0.999i)25-s + (−0.876 + 0.481i)27-s + (0.756 + 0.653i)29-s + (0.944 − 0.328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.892 + 0.450i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (4813, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.892 + 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001313180318 + 0.005517290974i\)
\(L(\frac12)\) \(\approx\) \(0.001313180318 + 0.005517290974i\)
\(L(1)\) \(\approx\) \(0.7116259738 - 0.08821416750i\)
\(L(1)\) \(\approx\) \(0.7116259738 - 0.08821416750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.985 + 0.166i)T \)
5 \( 1 + (0.699 - 0.714i)T \)
7 \( 1 + (-0.637 - 0.770i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.957 - 0.289i)T \)
17 \( 1 + (0.146 + 0.989i)T \)
19 \( 1 + (-0.604 + 0.796i)T \)
23 \( 1 + (0.387 - 0.921i)T \)
29 \( 1 + (0.756 + 0.653i)T \)
31 \( 1 + (0.944 - 0.328i)T \)
37 \( 1 + (0.0209 + 0.999i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.968 + 0.248i)T \)
47 \( 1 + (-0.699 + 0.714i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.463 - 0.886i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.756 + 0.653i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.604 + 0.796i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.268 - 0.963i)T \)
97 \( 1 + (0.985 + 0.166i)T \)
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

−17.67859050196974080219314972044, −16.71861617623243157947928580952, −16.189781256658538976569500362554, −15.45132127005541687606304385297, −15.17185347772257331929306168244, −13.825758341529816621148609613434, −13.410743459582861962327571054823, −13.04994658898507859554789478376, −11.86029839204380807198959995307, −11.66704391546611880427831580606, −10.671404697821885324697707705880, −10.39124390987374418612375735804, −9.507141194056480278538051167312, −8.93561671417351382537863851179, −7.95190768493463690930740812308, −7.07091705049167570236751721427, −6.47139771850041486895204653827, −6.01931616163521414162953626300, −5.315455323491693515131475278813, −4.75281073595373259168973381123, −3.512237089615433707562263365994, −2.82154628620052121508669774975, −2.14405961698413416341149436595, −1.153087834511758935653865667060, −0.0018129881993997771242144871, 1.096422528110749607643575496288, 1.607019001402600269944990616100, 2.80466914492893593467630573104, 3.77098674087580656035680160830, 4.467856742087784055786952532270, 5.07825440770987429256107251594, 5.8708825304413905788316418207, 6.41902135050219210460398178881, 6.920219150874076152883528023690, 8.20190014171965500398173358407, 8.43727952157069048499136743648, 9.70499781003614447529558893787, 10.18769735017472312493973506134, 10.48058947081465362118173162143, 11.25656303170962612396849925776, 12.355391474399151365369144870693, 12.69954260311116863074673319922, 13.2037630810703015824948442579, 13.82824359636446127251808723688, 14.835794453012103131602651558335, 15.643699827402588769087262715521, 16.14071495591444228500534466786, 16.8505208651569938993257405596, 17.12365406237967183932564179861, 17.85934075249486059390000868329

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