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Properties

Label 1-6003-6003.1019-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.121 - 0.992i$
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  + (0.976 − 0.215i)2-s + (0.906 − 0.421i)4-s + (−0.115 + 0.993i)5-s + (0.912 − 0.409i)7-s + (0.794 − 0.607i)8-s + (0.101 + 0.994i)10-s + (−0.155 − 0.987i)11-s + (−0.248 − 0.968i)13-s + (0.802 − 0.596i)14-s + (0.644 − 0.764i)16-s + (0.959 + 0.281i)17-s + (−0.818 − 0.574i)19-s + (0.314 + 0.949i)20-s + (−0.365 − 0.930i)22-s + (−0.973 − 0.229i)25-s + (−0.452 − 0.891i)26-s + ⋯
L(s)  = 1  + (0.976 − 0.215i)2-s + (0.906 − 0.421i)4-s + (−0.115 + 0.993i)5-s + (0.912 − 0.409i)7-s + (0.794 − 0.607i)8-s + (0.101 + 0.994i)10-s + (−0.155 − 0.987i)11-s + (−0.248 − 0.968i)13-s + (0.802 − 0.596i)14-s + (0.644 − 0.764i)16-s + (0.959 + 0.281i)17-s + (−0.818 − 0.574i)19-s + (0.314 + 0.949i)20-s + (−0.365 − 0.930i)22-s + (−0.973 − 0.229i)25-s + (−0.452 − 0.891i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.331373636 - 2.633266166i\)
\(L(\frac12)\) \(\approx\) \(2.331373636 - 2.633266166i\)
\(L(1)\) \(\approx\) \(1.923473209 - 0.5748580079i\)
\(L(1)\) \(\approx\) \(1.923473209 - 0.5748580079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.976 - 0.215i)T \)
5 \( 1 + (-0.115 + 0.993i)T \)
7 \( 1 + (0.912 - 0.409i)T \)
11 \( 1 + (-0.155 - 0.987i)T \)
13 \( 1 + (-0.248 - 0.968i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (-0.818 - 0.574i)T \)
31 \( 1 + (-0.855 - 0.517i)T \)
37 \( 1 + (-0.591 + 0.806i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (0.855 - 0.517i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (-0.992 + 0.122i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (-0.275 - 0.961i)T \)
67 \( 1 + (-0.155 + 0.987i)T \)
71 \( 1 + (-0.996 - 0.0815i)T \)
73 \( 1 + (-0.742 + 0.670i)T \)
79 \( 1 + (-0.984 + 0.175i)T \)
83 \( 1 + (-0.942 + 0.333i)T \)
89 \( 1 + (0.768 - 0.639i)T \)
97 \( 1 + (0.464 + 0.885i)T \)
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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

−17.586766664144287872643197467746, −17.164433530133774902202765425469, −16.36416905966670003666568170746, −16.01402909285167996996693693101, −15.0769280715720116928694823377, −14.58577244672012087578803226517, −14.130570144882443660326440419546, −13.231465274062572928418161973224, −12.48152246815292618996276264864, −12.203282031668230982312566263706, −11.605273466523732167644554577636, −10.81890482152760651074656925601, −9.96997173197153031301615756141, −9.07010889982044027723455811791, −8.476564343448371257529489535141, −7.56272099248155160003572492912, −7.3303570543030908117402505993, −6.14718960900859165938067091799, −5.55760223017190023329974418772, −4.806851850292268385548361370183, −4.45599384269396554221331344204, −3.74699628902897985748732199488, −2.6151732021762373639930351892, −1.803126571761576472980701439233, −1.407586473468252059759656062822, 0.54169388762889764687410416006, 1.59656401422430338956882516534, 2.43429871549118245751560117743, 3.12529307416817356725495757051, 3.73517809220273594009776413957, 4.43868404011238997577447542501, 5.50558452369313132365147543431, 5.7009603291222703404743905777, 6.70837944215896208759268163410, 7.37218634414797340222640310859, 7.91340702572847297533210403113, 8.701027567707915683722038865876, 10.01677272963023897747999548427, 10.50431469167376503098807412901, 10.97525592860519134997141806503, 11.52473735397250123107746143178, 12.26503393811747237810788043023, 13.05905391500308425337966984531, 13.67918219850240639223408183530, 14.39919385787279349095179630253, 14.643359574510629589929433064294, 15.45069350731007918300250950175, 15.8929195913889815409946156518, 17.04234402947127374090263727528, 17.30846069605510372541089783883

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