↑

Properties

Label 1-6008-6008.5605-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.460 + 0.887i$
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  + (−0.728 + 0.684i)3-s + (0.425 + 0.904i)5-s + (0.728 − 0.684i)7-s + (0.0627 − 0.998i)9-s + (−0.309 − 0.951i)11-s + (−0.968 − 0.248i)13-s + (−0.929 − 0.368i)15-s + (−0.992 − 0.125i)17-s + (−0.535 + 0.844i)19-s + (−0.0627 + 0.998i)21-s + (0.535 + 0.844i)23-s + (−0.637 + 0.770i)25-s + (0.637 + 0.770i)27-s + (−0.0627 + 0.998i)29-s + (0.0627 − 0.998i)31-s + ⋯
L(s)  = 1  + (−0.728 + 0.684i)3-s + (0.425 + 0.904i)5-s + (0.728 − 0.684i)7-s + (0.0627 − 0.998i)9-s + (−0.309 − 0.951i)11-s + (−0.968 − 0.248i)13-s + (−0.929 − 0.368i)15-s + (−0.992 − 0.125i)17-s + (−0.535 + 0.844i)19-s + (−0.0627 + 0.998i)21-s + (0.535 + 0.844i)23-s + (−0.637 + 0.770i)25-s + (0.637 + 0.770i)27-s + (−0.0627 + 0.998i)29-s + (0.0627 − 0.998i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (5605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012680810 + 0.6151140473i\)
\(L(\frac12)\) \(\approx\) \(1.012680810 + 0.6151140473i\)
\(L(1)\) \(\approx\) \(0.8381653383 + 0.2134447883i\)
\(L(1)\) \(\approx\) \(0.8381653383 + 0.2134447883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.728 + 0.684i)T \)
5 \( 1 + (0.425 + 0.904i)T \)
7 \( 1 + (0.728 - 0.684i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.968 - 0.248i)T \)
17 \( 1 + (-0.992 - 0.125i)T \)
19 \( 1 + (-0.535 + 0.844i)T \)
23 \( 1 + (0.535 + 0.844i)T \)
29 \( 1 + (-0.0627 + 0.998i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (0.637 - 0.770i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.425 - 0.904i)T \)
47 \( 1 + (-0.425 - 0.904i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.187 - 0.982i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.0627 + 0.998i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.425 - 0.904i)T \)
97 \( 1 + (0.728 + 0.684i)T \)
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

−17.57280739052929796608798934673, −17.228634859414702593257991826523, −16.4515718144579519920095357181, −15.666392791555384384993573467009, −15.04048155550399403025260646592, −14.29173016441483241224390100696, −13.43339069783584134417380641470, −12.84613377731526049463582053088, −12.38919984349847158274139600334, −11.83826618987268506484851462812, −11.09078024773398135048659418210, −10.433817201114686378266326330908, −9.49965627635609843837524104866, −8.94739846466020573032847715064, −8.13891730667283525293395427789, −7.58351464980345198669480466970, −6.6464272307688632228057265087, −6.20003086005631274533116089841, −5.15938111083580301070254149053, −4.742822718071942307902315195532, −4.46255855522188712125962315925, −2.54871514944706964511479358504, −2.23073574019340014114963138252, −1.52508665942945561878143425845, −0.48431776407032699743961980739, 0.642967996268106162079703900098, 1.75065258945707889482014756553, 2.64995300917431665183497816391, 3.4860872669782134924251143489, 4.15361777996682906070361293112, 4.94927602794141227274290165754, 5.63093048652837829360145284857, 6.198876952908969195210067956589, 7.07642842642000730295904662294, 7.56499592593019898321248324674, 8.54154310096121297446366908001, 9.36366125689344590467450649361, 10.0825706350759669348626786340, 10.59308778181708344978287219605, 11.21274459805888823179927841476, 11.46577501764457123937621245549, 12.54739460845735339201054017609, 13.31357194373507060711484736437, 13.9983608952945572270572759459, 14.75389508395427370984908915223, 15.032669440042650205369865077227, 15.9063597323545848200641746460, 16.706848913907237485564772505318, 17.12264658822452683147919759298, 17.77470390693947007093763789520

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