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Properties

Label 1-6003-6003.3004-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.928 + 0.370i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.458 + 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (0.327 − 0.945i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (0.998 + 0.0475i)11-s + (0.327 + 0.945i)13-s + (0.690 + 0.723i)14-s + (−0.327 + 0.945i)16-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (−0.989 − 0.142i)26-s + ⋯
L(s)  = 1  + (−0.458 + 0.888i)2-s + (−0.580 − 0.814i)4-s + (0.235 − 0.971i)5-s + (0.327 − 0.945i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (0.998 + 0.0475i)11-s + (0.327 + 0.945i)13-s + (0.690 + 0.723i)14-s + (−0.327 + 0.945i)16-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (−0.989 − 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.928 + 0.370i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (3004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.928 + 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.637139442 + 0.3143439616i\)
\(L(\frac12)\) \(\approx\) \(1.637139442 + 0.3143439616i\)
\(L(1)\) \(\approx\) \(0.9784285927 + 0.1560701613i\)
\(L(1)\) \(\approx\) \(0.9784285927 + 0.1560701613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.458 + 0.888i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.327 - 0.945i)T \)
11 \( 1 + (0.998 + 0.0475i)T \)
13 \( 1 + (0.327 + 0.945i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (0.371 - 0.928i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (0.971 + 0.235i)T \)
43 \( 1 + (0.371 + 0.928i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (-0.618 - 0.786i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.189 + 0.981i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (0.690 - 0.723i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.85696350010097459848284005371, −17.51152069365830844763139537950, −16.44548180805131814607798484027, −15.62474404336107543292238280690, −15.137172117721335271836199530655, −14.119281138453033566777270972374, −13.86361931598541039527013384364, −12.94669944307607109201829878952, −12.145761106698409965081231680277, −11.71584850932109746146068780158, −10.94972170308982288574963973869, −10.60605129881302773644519180239, −9.669658844668693150874190037636, −9.04662645541704986229311443451, −8.65319870278641185894119478352, −7.59776571934184866480626674784, −7.117004604163291863664105372330, −6.1432055858139419767615478012, −5.455621314057023060996566909244, −4.558652191072797879180462246110, −3.66282827290397505533320121180, −2.99454182322265286048491324181, −2.4200521098228268657818952019, −1.680559005357386259979386697292, −0.692735832900689235206028522718, 0.80882439531369515932147422930, 1.3325636355535708302478467072, 2.047602780550505046393274549963, 3.68694525733057496314563074853, 4.398484726633330427599247376509, 4.6313956250231595941120510531, 5.76308216213688206579046038755, 6.28707023967517029045540715808, 7.0260843200925758742920460164, 7.69331256678808260314750717266, 8.44455222845123124302053395871, 8.99469594551195567577039770838, 9.61703465523690493960094521181, 10.15821683684446315861544438969, 11.1992131488867710836028916065, 11.63346025147648925551184884010, 12.704618948194942203609435857283, 13.38880921817347695987356327469, 14.02597216955392934298532443483, 14.29448257253619018804154528375, 15.32265820193243284817123956412, 15.93327752964788440858299456209, 16.6480649102185643576454939753, 17.02394709951949775152031882301, 17.44903409376565746444276050866

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