Properties

Label 1-6003-6003.689-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.927 + 0.372i$
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  + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.826 + 0.563i)5-s + (0.988 − 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (−0.0747 − 0.997i)14-s + (0.955 + 0.294i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.733 − 0.680i)20-s + (−0.365 + 0.930i)22-s + (0.365 + 0.930i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.826 + 0.563i)5-s + (0.988 − 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.623 − 0.781i)10-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (−0.0747 − 0.997i)14-s + (0.955 + 0.294i)16-s − 17-s + (0.623 − 0.781i)19-s + (−0.733 − 0.680i)20-s + (−0.365 + 0.930i)22-s + (0.365 + 0.930i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.927 + 0.372i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.927 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394530063 + 0.2696905915i\)
\(L(\frac12)\) \(\approx\) \(1.394530063 + 0.2696905915i\)
\(L(1)\) \(\approx\) \(1.021195421 - 0.3232514368i\)
\(L(1)\) \(\approx\) \(1.021195421 - 0.3232514368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.0747 - 0.997i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.826 - 0.563i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (-0.955 + 0.294i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.733 - 0.680i)T \)
83 \( 1 + (0.365 + 0.930i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.365 + 0.930i)T \)
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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

−17.52882375547897814422464841548, −17.19171175794126507400461477622, −16.287305872492072474266674648137, −15.77945160558198264997437473553, −15.0328148280057256522293210936, −14.45094693472030964618437382351, −13.88787091505871058798823134933, −13.15662786557734746414820246310, −12.63258873670637611487650859639, −12.03601972410102622171214513178, −10.82648379057927751114785582297, −10.30847859230711565928662819756, −9.44566916484480331183614211627, −8.98541472894944185389643243655, −8.06514899818669172154552679644, −7.765555506361905997568287284683, −6.95432621099809389357468321724, −5.98782974533667837650415905472, −5.42938109987998282739259137413, −4.92176410954058811309333703558, −4.41832731590685513851072411683, −3.25236790513750226294027541039, −2.26322245552144139329625299518, −1.534589599205617986001804395280, −0.37346515289443799456105475082, 0.9378657391552326313549986211, 1.94512667902377117111467401919, 2.34314393442689524117317980644, 3.005845840768558340122232053456, 4.04579272446075214824462547122, 4.80739131975154495938745878402, 5.28698304989038820731420556314, 6.0501125866580609088708457610, 7.15330198615529972088089738673, 7.64820991583120303850615158820, 8.76176491315663515289701423236, 9.11336959312428472264066291402, 9.98212371426498924737920583785, 10.58804267289111811644163534830, 11.10116598879893165196527912994, 11.62579246122459581896985121682, 12.45074235169579308014235953201, 13.305334799814905121366796526649, 13.73651287192926990978766494464, 14.20419262544790603373689619654, 14.98242044342891991598416362704, 15.56675621654109798261084663311, 16.75867396166318644241688983272, 17.470606261373113367309397603281, 17.72198716107686095177971768812

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