↑

Properties

Label 1-6008-6008.613-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.911 + 0.410i$
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  + (−0.0961 + 0.995i)3-s + (0.220 − 0.975i)5-s + (0.112 + 0.993i)7-s + (−0.981 − 0.191i)9-s + (−0.228 − 0.973i)11-s + (−0.244 + 0.969i)13-s + (0.949 + 0.312i)15-s + (0.276 − 0.960i)17-s + (−0.773 − 0.634i)19-s + (−0.999 + 0.0167i)21-s + (0.549 + 0.835i)23-s + (−0.903 − 0.429i)25-s + (0.285 − 0.958i)27-s + (0.818 + 0.574i)29-s + (−0.485 + 0.874i)31-s + ⋯
L(s)  = 1  + (−0.0961 + 0.995i)3-s + (0.220 − 0.975i)5-s + (0.112 + 0.993i)7-s + (−0.981 − 0.191i)9-s + (−0.228 − 0.973i)11-s + (−0.244 + 0.969i)13-s + (0.949 + 0.312i)15-s + (0.276 − 0.960i)17-s + (−0.773 − 0.634i)19-s + (−0.999 + 0.0167i)21-s + (0.549 + 0.835i)23-s + (−0.903 − 0.429i)25-s + (0.285 − 0.958i)27-s + (0.818 + 0.574i)29-s + (−0.485 + 0.874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.911 + 0.410i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.911 + 0.410i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1445062015 + 0.6735307647i\)
\(L(\frac12)\) \(\approx\) \(0.1445062015 + 0.6735307647i\)
\(L(1)\) \(\approx\) \(0.8543895813 + 0.2330511839i\)
\(L(1)\) \(\approx\) \(0.8543895813 + 0.2330511839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.0961 + 0.995i)T \)
5 \( 1 + (0.220 - 0.975i)T \)
7 \( 1 + (0.112 + 0.993i)T \)
11 \( 1 + (-0.228 - 0.973i)T \)
13 \( 1 + (-0.244 + 0.969i)T \)
17 \( 1 + (0.276 - 0.960i)T \)
19 \( 1 + (-0.773 - 0.634i)T \)
23 \( 1 + (0.549 + 0.835i)T \)
29 \( 1 + (0.818 + 0.574i)T \)
31 \( 1 + (-0.485 + 0.874i)T \)
37 \( 1 + (-0.478 - 0.878i)T \)
41 \( 1 + (0.728 - 0.684i)T \)
43 \( 1 + (-0.0125 - 0.999i)T \)
47 \( 1 + (-0.995 + 0.0920i)T \)
53 \( 1 + (-0.876 + 0.481i)T \)
59 \( 1 + (0.154 + 0.988i)T \)
61 \( 1 + (0.855 + 0.518i)T \)
67 \( 1 + (0.799 - 0.601i)T \)
71 \( 1 + (0.212 + 0.977i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.773 - 0.634i)T \)
83 \( 1 + (-0.146 - 0.989i)T \)
89 \( 1 + (-0.872 + 0.489i)T \)
97 \( 1 + (-0.916 + 0.399i)T \)
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

−17.410172511252398576773314382219, −17.16500831003616005979912023746, −16.29085734374726705834863768897, −15.051958415677228480801636830262, −14.83445013216404321782504431787, −14.189651327961742209257107650563, −13.42891423265425690458069625223, −12.70635909115146091678346375580, −12.524523531806233329599820888123, −11.271734428069099705933246889129, −10.95143956387140651929394018507, −10.04871339097088755443510983751, −9.851948850261122600898933117417, −8.29677261462800957224632284176, −7.974606290277494971271895197259, −7.36017487513363612186250139645, −6.486464160753396811505013785230, −6.3473984953063736314291772011, −5.2714917565915313370010852953, −4.43316546187499499870192610636, −3.512150862677196127288964748707, −2.7777530083768009429758988807, −2.02831606799930754214071148527, −1.33023603709455577567327377756, −0.18807835150375903047075930806, 0.98024841138168000535031940302, 2.096273035719669115974490594649, 2.81499954740207739486035263245, 3.63405230530036228177896495988, 4.49438463121311939596731419983, 5.20527929801797082891735326736, 5.44500192974180212958374986165, 6.311986292117414392872010855824, 7.23507328198813645728871903433, 8.39726376036819458208119100116, 8.825586281022878942335468202003, 9.191155165074317478835679121499, 9.84676426324604260494039079045, 10.83627965287506922766180675685, 11.351563347987433411094413079131, 12.028745877066252073033533883103, 12.58103433124124935003379484277, 13.53959011607490633045115307640, 14.123028889833395684728201346004, 14.73535846754099136203396159066, 15.69358864315877116535543637621, 16.031173455139693293252035353854, 16.44731024174523197059749357379, 17.3318913756195974934832420912, 17.72469678311746360386344180718

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