↑

Properties

Label 1-6008-6008.61-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.619 - 0.784i$
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  + (0.0209 + 0.999i)3-s + (0.957 + 0.289i)5-s + (0.876 + 0.481i)7-s + (−0.999 + 0.0418i)9-s + (−0.669 + 0.743i)11-s + (0.348 − 0.937i)13-s + (−0.268 + 0.963i)15-s + (−0.570 − 0.821i)17-s + (−0.783 − 0.621i)19-s + (−0.463 + 0.886i)21-s + (0.146 + 0.989i)23-s + (0.832 + 0.553i)25-s + (−0.0627 − 0.998i)27-s + (−0.463 + 0.886i)29-s + (−0.999 + 0.0418i)31-s + ⋯
L(s)  = 1  + (0.0209 + 0.999i)3-s + (0.957 + 0.289i)5-s + (0.876 + 0.481i)7-s + (−0.999 + 0.0418i)9-s + (−0.669 + 0.743i)11-s + (0.348 − 0.937i)13-s + (−0.268 + 0.963i)15-s + (−0.570 − 0.821i)17-s + (−0.783 − 0.621i)19-s + (−0.463 + 0.886i)21-s + (0.146 + 0.989i)23-s + (0.832 + 0.553i)25-s + (−0.0627 − 0.998i)27-s + (−0.463 + 0.886i)29-s + (−0.999 + 0.0418i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.619 - 0.784i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.619 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2022927926 + 0.4174921868i\)
\(L(\frac12)\) \(\approx\) \(-0.2022927926 + 0.4174921868i\)
\(L(1)\) \(\approx\) \(0.9210081714 + 0.4643589607i\)
\(L(1)\) \(\approx\) \(0.9210081714 + 0.4643589607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.0209 + 0.999i)T \)
5 \( 1 + (0.957 + 0.289i)T \)
7 \( 1 + (0.876 + 0.481i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.348 - 0.937i)T \)
17 \( 1 + (-0.570 - 0.821i)T \)
19 \( 1 + (-0.783 - 0.621i)T \)
23 \( 1 + (0.146 + 0.989i)T \)
29 \( 1 + (-0.463 + 0.886i)T \)
31 \( 1 + (-0.999 + 0.0418i)T \)
37 \( 1 + (-0.832 - 0.553i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.728 - 0.684i)T \)
47 \( 1 + (-0.957 - 0.289i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.604 - 0.796i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.463 + 0.886i)T \)
71 \( 1 + (0.968 + 0.248i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.783 - 0.621i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.228 + 0.973i)T \)
97 \( 1 + (-0.0209 + 0.999i)T \)
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

−17.288429710002127223790824827695, −16.805850990676746247488683956925, −16.333956695011479854736506677358, −15.06745047408225608958173265834, −14.47755670906862691619389265961, −13.893828346317595996308325642132, −13.3749241432627869571701042248, −12.86576014872043027874817488821, −12.150127933427113227989204324296, −11.21783048475255164255938810656, −10.83913253048573157010489586372, −10.11367786281048757336660450858, −9.00342859036873046501568810746, −8.51742937623141841529548762322, −8.04215784080198043603609184658, −7.14233196359011925210093470765, −6.34475694774031579216265366121, −5.99018858280183326883497654884, −5.12455524336034035534114584327, −4.38000646932438022902461914962, −3.445120585285814827694072302732, −2.322392360983904794834021622298, −1.854328549929620187740366976609, −1.28699084483032148559069109445, −0.09848230101515210349798589490, 1.5166304902229636337480697077, 2.247859856539717859164370389, 2.89491462564758040692989146525, 3.67677487868297575090000663131, 4.881171702103747821984086036201, 5.06876565032258996945349806411, 5.67685849774049055735753449730, 6.582209700658906915747984337917, 7.471167160929830211997828173903, 8.21693767444520714440016328414, 9.11318460092503692345088964667, 9.34555693935958465584444626486, 10.27392504694103037654402281255, 10.882112293370737050365064393023, 11.14809820347496590107256298740, 12.18895884754648520493494979009, 13.01990612856620605591662456803, 13.58509836379889479753757520987, 14.38417617934917161921079655096, 14.9870445723543933470401143292, 15.3799235683066209894534526032, 16.02077617325221709485809132373, 16.96010429960651774098623519254, 17.53115824163126848807509210813, 18.04690589963740910124322136920

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