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Properties

Label 1-4009-4009.1068-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.992 + 0.118i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (0.289 − 0.957i)2-s + (0.905 − 0.424i)3-s + (−0.831 − 0.555i)4-s + (−0.261 + 0.965i)5-s + (−0.144 − 0.989i)6-s + (0.992 − 0.119i)7-s + (−0.772 + 0.635i)8-s + (0.638 − 0.769i)9-s + (0.848 + 0.529i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (−0.299 + 0.954i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (0.383 + 0.923i)16-s + (−0.909 − 0.415i)17-s + ⋯
L(s)  = 1  + (0.289 − 0.957i)2-s + (0.905 − 0.424i)3-s + (−0.831 − 0.555i)4-s + (−0.261 + 0.965i)5-s + (−0.144 − 0.989i)6-s + (0.992 − 0.119i)7-s + (−0.772 + 0.635i)8-s + (0.638 − 0.769i)9-s + (0.848 + 0.529i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (−0.299 + 0.954i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (0.383 + 0.923i)16-s + (−0.909 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1068, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.992 + 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07466404610 - 1.257991814i\)
\(L(\frac12)\) \(\approx\) \(-0.07466404610 - 1.257991814i\)
\(L(1)\) \(\approx\) \(1.026084022 - 0.6992427046i\)
\(L(1)\) \(\approx\) \(1.026084022 - 0.6992427046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.289 - 0.957i)T \)
3 \( 1 + (0.905 - 0.424i)T \)
5 \( 1 + (-0.261 + 0.965i)T \)
7 \( 1 + (0.992 - 0.119i)T \)
11 \( 1 + (-0.873 + 0.486i)T \)
13 \( 1 + (-0.299 + 0.954i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
23 \( 1 + (-0.719 - 0.694i)T \)
29 \( 1 + (-0.676 - 0.736i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (0.858 - 0.512i)T \)
41 \( 1 + (0.803 + 0.595i)T \)
43 \( 1 + (-0.661 - 0.749i)T \)
47 \( 1 + (0.868 - 0.495i)T \)
53 \( 1 + (0.591 - 0.806i)T \)
59 \( 1 + (0.114 - 0.993i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (-0.797 - 0.603i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (0.995 + 0.0995i)T \)
79 \( 1 + (-0.759 + 0.650i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.932 + 0.360i)T \)
97 \( 1 + (0.739 - 0.672i)T \)
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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

−18.60044156474243564969435670255, −18.08533874909282878426856382316, −17.33505481477457333660997309810, −16.60600561717535197682712071445, −15.93820404865266049860485660482, −15.377808379550685027758588347645, −14.9570653893951068080773758168, −14.15601290004709321488731739437, −13.4331251622506322359069646019, −12.97796849077545731079646861478, −12.31830886768529636957927882260, −11.221921519074089149607982501314, −10.47033470546801735127703771369, −9.475144902825717892167249825091, −8.84649148230065604424064653033, −8.3162755893474868011363464006, −7.7317913772443858635256828880, −7.35902580103025253134098518915, −5.79541596941932544065518905053, −5.39940291043277177019179102824, −4.61796549656129169108433328706, −4.10519548476828489187510042450, −3.22835702207606050550892056773, −2.29452998655526433077512602438, −1.2064348252174844899741810630, 0.26459317876827042049618079819, 1.712745623991751027114105514522, 2.30248775865472967587212753433, 2.58806542369952186087039391216, 3.87080147947783939705935863863, 4.18319479809251206846093998804, 5.08322893322126764258072961294, 6.16894119747121217380837310688, 7.07263290193085924993903294821, 7.6940828536427042815348160161, 8.3796005102796526745072587243, 9.22733078283646997694947404135, 9.871286427298624702632982722056, 10.65668639874389895188047611421, 11.2946602340681344776909148384, 11.86984345885204281906072423966, 12.67641488915222649529860109717, 13.42711358311208053142587352629, 14.02979964748164441564425672062, 14.5361645520720025431761129554, 15.07512818587575157883140392842, 15.69732620379370879053063528880, 17.01391891438013308225780326662, 18.029361197138430524834464087639, 18.349320069164306734714104495489

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