↑

Properties

Label 1-6008-6008.549-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.645 + 0.763i$
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  + (0.00418 + 0.999i)3-s + (0.998 + 0.0586i)5-s + (0.402 − 0.915i)7-s + (−0.999 + 0.00837i)9-s + (−0.146 + 0.989i)11-s + (0.528 + 0.848i)13-s + (−0.0544 + 0.998i)15-s + (−0.981 − 0.191i)17-s + (−0.433 + 0.901i)19-s + (0.916 + 0.399i)21-s + (−0.611 − 0.791i)23-s + (0.993 + 0.117i)25-s + (−0.0125 − 0.999i)27-s + (0.662 − 0.748i)29-s + (0.804 − 0.594i)31-s + ⋯
L(s)  = 1  + (0.00418 + 0.999i)3-s + (0.998 + 0.0586i)5-s + (0.402 − 0.915i)7-s + (−0.999 + 0.00837i)9-s + (−0.146 + 0.989i)11-s + (0.528 + 0.848i)13-s + (−0.0544 + 0.998i)15-s + (−0.981 − 0.191i)17-s + (−0.433 + 0.901i)19-s + (0.916 + 0.399i)21-s + (−0.611 − 0.791i)23-s + (0.993 + 0.117i)25-s + (−0.0125 − 0.999i)27-s + (0.662 − 0.748i)29-s + (0.804 − 0.594i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.645 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7615128033 + 1.641070570i\)
\(L(\frac12)\) \(\approx\) \(0.7615128033 + 1.641070570i\)
\(L(1)\) \(\approx\) \(1.097888769 + 0.4969023605i\)
\(L(1)\) \(\approx\) \(1.097888769 + 0.4969023605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.00418 + 0.999i)T \)
5 \( 1 + (0.998 + 0.0586i)T \)
7 \( 1 + (0.402 - 0.915i)T \)
11 \( 1 + (-0.146 + 0.989i)T \)
13 \( 1 + (0.528 + 0.848i)T \)
17 \( 1 + (-0.981 - 0.191i)T \)
19 \( 1 + (-0.433 + 0.901i)T \)
23 \( 1 + (-0.611 - 0.791i)T \)
29 \( 1 + (0.662 - 0.748i)T \)
31 \( 1 + (0.804 - 0.594i)T \)
37 \( 1 + (-0.418 + 0.908i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (0.888 - 0.459i)T \)
47 \( 1 + (0.773 + 0.634i)T \)
53 \( 1 + (-0.968 + 0.248i)T \)
59 \( 1 + (-0.129 - 0.991i)T \)
61 \( 1 + (0.699 + 0.714i)T \)
67 \( 1 + (-0.0961 + 0.995i)T \)
71 \( 1 + (0.260 + 0.965i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.433 + 0.901i)T \)
83 \( 1 + (-0.944 + 0.328i)T \)
89 \( 1 + (0.0460 + 0.998i)T \)
97 \( 1 + (0.591 - 0.806i)T \)
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

−17.53922512348974369645075192646, −17.25162609855610152259294801458, −16.06679446279028978604078458209, −15.576510459281045616238796261035, −14.73783979917948859236713175154, −13.91751335604062380235256511593, −13.60458748618175599524151204486, −12.92230029867195585705810739016, −12.3699796546889901640925135235, −11.56244308340192040444141286742, −10.89696955279692158325046312720, −10.385410521625297963439284701841, −9.07882311166104563327315524726, −8.82153414453303311538486798887, −8.24876598421112837324189829873, −7.39390512714687587216235409486, −6.379542177395272900820016231860, −6.13656943464090713489999060059, −5.39266370217143853777340081826, −4.82937669087673854835427620913, −3.38615380740391496240746466294, −2.75998551624695647854363183450, −2.07526025446186520674279253208, −1.430259907447000269384337978735, −0.44083099028223728646943592901, 1.097477561734027373777856714670, 2.062866474343943765378337883218, 2.573157194532495674870415869456, 3.77741154758161712170307511647, 4.44894556401250707770359450502, 4.688572181497496634374249618103, 5.770521422831762593812399514447, 6.43224867564716008260582808109, 7.00427601402695620920278680933, 8.16846510743689768084274542913, 8.61487866624035061068609708628, 9.57210605169646036962353356195, 10.01070531990451449271814487761, 10.47059894924822162369127888523, 11.184953778175497721476062614630, 11.86511275669130643650286744077, 12.793314878834283480753402334465, 13.60504805004559925119568998112, 14.11681652902936806930831481288, 14.511961035985447034831496681164, 15.4281528839238368108951556350, 15.93588510839313635015199176116, 16.84627052312440704671900818909, 17.18138324792646447803435478949, 17.693354466795183788414917559808

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