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Properties

Label 1-6003-6003.119-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.290 + 0.956i$
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  + (−0.255 − 0.966i)2-s + (−0.869 + 0.494i)4-s + (0.938 − 0.346i)5-s + (−0.0882 + 0.996i)7-s + (0.699 + 0.714i)8-s + (−0.574 − 0.818i)10-s + (−0.108 + 0.994i)11-s + (0.833 + 0.552i)13-s + (0.985 − 0.169i)14-s + (0.511 − 0.859i)16-s + (−0.989 − 0.142i)17-s + (−0.505 + 0.862i)19-s + (−0.644 + 0.764i)20-s + (0.988 − 0.149i)22-s + (0.760 − 0.649i)25-s + (0.320 − 0.947i)26-s + ⋯
L(s)  = 1  + (−0.255 − 0.966i)2-s + (−0.869 + 0.494i)4-s + (0.938 − 0.346i)5-s + (−0.0882 + 0.996i)7-s + (0.699 + 0.714i)8-s + (−0.574 − 0.818i)10-s + (−0.108 + 0.994i)11-s + (0.833 + 0.552i)13-s + (0.985 − 0.169i)14-s + (0.511 − 0.859i)16-s + (−0.989 − 0.142i)17-s + (−0.505 + 0.862i)19-s + (−0.644 + 0.764i)20-s + (0.988 − 0.149i)22-s + (0.760 − 0.649i)25-s + (0.320 − 0.947i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4969954788 + 0.6702375350i\)
\(L(\frac12)\) \(\approx\) \(0.4969954788 + 0.6702375350i\)
\(L(1)\) \(\approx\) \(0.8779581835 - 0.1112951298i\)
\(L(1)\) \(\approx\) \(0.8779581835 - 0.1112951298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.255 - 0.966i)T \)
5 \( 1 + (0.938 - 0.346i)T \)
7 \( 1 + (-0.0882 + 0.996i)T \)
11 \( 1 + (-0.108 + 0.994i)T \)
13 \( 1 + (0.833 + 0.552i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (-0.505 + 0.862i)T \)
31 \( 1 + (-0.798 - 0.601i)T \)
37 \( 1 + (-0.607 + 0.794i)T \)
41 \( 1 + (-0.0950 + 0.995i)T \)
43 \( 1 + (0.798 - 0.601i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (-0.742 + 0.670i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (-0.585 - 0.810i)T \)
67 \( 1 + (-0.994 + 0.108i)T \)
71 \( 1 + (0.882 + 0.470i)T \)
73 \( 1 + (-0.953 - 0.301i)T \)
79 \( 1 + (0.0135 + 0.999i)T \)
83 \( 1 + (0.546 - 0.837i)T \)
89 \( 1 + (-0.852 - 0.523i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−17.485028051774938411068077623349, −16.967825854891087070785856723265, −16.18406725050586877630647371907, −15.72426492825505458089253609109, −14.91416274158342483592928127572, −14.16021508100610195714435037245, −13.68306835316377163496198346863, −13.25638649044051946783575349356, −12.682061581696728308734289517885, −11.12829429472546334791004289802, −10.69157107167074026802167894578, −10.37407640838382476530019624916, −9.201872236197343768843337063536, −8.948410570493586363477176133194, −8.12962861876635540801341627342, −7.275682491679756201229956134812, −6.7115292134052867605441120890, −6.117153420343912193072641354055, −5.51907486451164891898021175019, −4.71851842976757623140013656182, −3.85179995097515144932156619053, −3.17444084825465237697583741783, −2.01839054963378066906208618563, −1.10589788820935992318851239146, −0.23172671964260529454832935090, 1.35157355237389587647194097401, 1.87488009571205401650114221906, 2.40459200421421162074373924554, 3.25112526067535770133994632895, 4.34087857259177613805609065188, 4.70954936767783356567911374147, 5.73032893620389673037227848587, 6.20848722836024859724543495435, 7.220590136792296698185320253266, 8.22323103334937294634482303828, 8.829387583837818551320701631061, 9.32148025805356481963506756386, 9.840035349852224851840103567417, 10.65708301003416625457698290209, 11.2293015356880331302509446630, 12.14060944652926235348038807850, 12.52493047919909449629455745795, 13.1664101819869701300898499919, 13.7669343477490860164914061383, 14.48007039592828751035379800995, 15.26029514350191535215177356345, 16.040945996424625995946718704047, 16.860560076311096666023410820637, 17.4010509691296136123934602765, 18.096648413357070064175654134624

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