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Properties

Label 1-4003-4003.214-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.524 - 0.851i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (−0.302 − 0.953i)2-s + (−0.436 − 0.899i)3-s + (−0.817 + 0.576i)4-s + (−0.773 + 0.633i)5-s + (−0.725 + 0.687i)6-s + (−0.0180 + 0.999i)7-s + (0.796 + 0.605i)8-s + (−0.619 + 0.785i)9-s + (0.837 + 0.546i)10-s + (0.647 + 0.762i)11-s + (0.874 + 0.484i)12-s + (0.267 + 0.963i)13-s + (0.958 − 0.284i)14-s + (0.907 + 0.419i)15-s + (0.336 − 0.941i)16-s + (−0.994 − 0.108i)17-s + ⋯
L(s)  = 1  + (−0.302 − 0.953i)2-s + (−0.436 − 0.899i)3-s + (−0.817 + 0.576i)4-s + (−0.773 + 0.633i)5-s + (−0.725 + 0.687i)6-s + (−0.0180 + 0.999i)7-s + (0.796 + 0.605i)8-s + (−0.619 + 0.785i)9-s + (0.837 + 0.546i)10-s + (0.647 + 0.762i)11-s + (0.874 + 0.484i)12-s + (0.267 + 0.963i)13-s + (0.958 − 0.284i)14-s + (0.907 + 0.419i)15-s + (0.336 − 0.941i)16-s + (−0.994 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7701209934 - 0.4300750774i\)
\(L(\frac12)\) \(\approx\) \(0.7701209934 - 0.4300750774i\)
\(L(1)\) \(\approx\) \(0.6284942285 - 0.2448120346i\)
\(L(1)\) \(\approx\) \(0.6284942285 - 0.2448120346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.302 - 0.953i)T \)
3 \( 1 + (-0.436 - 0.899i)T \)
5 \( 1 + (-0.773 + 0.633i)T \)
7 \( 1 + (-0.0180 + 0.999i)T \)
11 \( 1 + (0.647 + 0.762i)T \)
13 \( 1 + (0.267 + 0.963i)T \)
17 \( 1 + (-0.994 - 0.108i)T \)
19 \( 1 + (0.403 + 0.915i)T \)
23 \( 1 + (0.336 - 0.941i)T \)
29 \( 1 + (0.403 - 0.915i)T \)
31 \( 1 + (-0.0901 - 0.995i)T \)
37 \( 1 + (0.403 - 0.915i)T \)
41 \( 1 + (0.647 - 0.762i)T \)
43 \( 1 + (0.997 - 0.0721i)T \)
47 \( 1 + (0.336 + 0.941i)T \)
53 \( 1 + (0.336 + 0.941i)T \)
59 \( 1 + (-0.922 + 0.386i)T \)
61 \( 1 + (-0.232 + 0.972i)T \)
67 \( 1 + (0.958 + 0.284i)T \)
71 \( 1 + (-0.436 - 0.899i)T \)
73 \( 1 + (-0.947 - 0.319i)T \)
79 \( 1 + (-0.232 - 0.972i)T \)
83 \( 1 + (-0.302 - 0.953i)T \)
89 \( 1 + (0.958 - 0.284i)T \)
97 \( 1 + (0.958 + 0.284i)T \)
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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

−18.28875481859152902661461477243, −17.53537757354362110658720496490, −17.156540872172821731616658717106, −16.48859232546821317708678905572, −15.820892768874835466333116728819, −15.58527571442060903118707837960, −14.73611603633948321286696570783, −13.97173774288613754257356961790, −13.26982967041176925827664021097, −12.569910779426687642309173785997, −11.23805643726576237332649684869, −11.1538248215787597987484815453, −10.21791718667076061306509761650, −9.43384152173007943002320804365, −8.76536476628767494758497518751, −8.29031503377845515281921851693, −7.324668254925863128804860992947, −6.70472331736611844920803560008, −5.86819555591040033355244417707, −4.98413702863529656914668186451, −4.626442984818822286559020251263, −3.70792686405254182980328755291, −3.26872658654210834029493554318, −1.10913720471118600636957571962, −0.70072407775895764881037901659, 0.53342158539221158598253903746, 1.717543045521794423389138637876, 2.30346179953515383951066418965, 2.90571744338800615885065019511, 4.22003079322581963179411643970, 4.43293638730188864857376496738, 5.82665445128842409211098181181, 6.405622845866390828120409890869, 7.377240227962619618047936036398, 7.79316420267177280050570113540, 8.86671273905067030828367083471, 9.15282792836616028092421551112, 10.30970008040315197210497319219, 11.04878415130415866570895291999, 11.59727514526617285840101628788, 12.12286756263501716752719194339, 12.49454244693840160469838753599, 13.39329204589633097604680973794, 14.244084612101216664839415054785, 14.70928459224930192034774959631, 15.78498068881491562905168819755, 16.467645322609446571481559640789, 17.37002728898536412557014233161, 17.89177427543243995446906577692, 18.671472264059351371930665515927

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