↑

Properties

Label 1-4009-4009.1010-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.573 - 0.819i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0647 + 0.997i)2-s + (−0.985 − 0.168i)3-s + (−0.991 − 0.129i)4-s + (−0.863 − 0.504i)5-s + (0.232 − 0.972i)6-s + (0.691 + 0.722i)7-s + (0.193 − 0.981i)8-s + (0.943 + 0.332i)9-s + (0.559 − 0.829i)10-s + (0.525 − 0.850i)11-s + (0.955 + 0.294i)12-s + (0.820 + 0.571i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + (0.966 + 0.256i)16-s + (−0.327 + 0.944i)17-s + ⋯
L(s)  = 1  + (−0.0647 + 0.997i)2-s + (−0.985 − 0.168i)3-s + (−0.991 − 0.129i)4-s + (−0.863 − 0.504i)5-s + (0.232 − 0.972i)6-s + (0.691 + 0.722i)7-s + (0.193 − 0.981i)8-s + (0.943 + 0.332i)9-s + (0.559 − 0.829i)10-s + (0.525 − 0.850i)11-s + (0.955 + 0.294i)12-s + (0.820 + 0.571i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + (0.966 + 0.256i)16-s + (−0.327 + 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02629133951 - 0.05047233601i\)
\(L(\frac12)\) \(\approx\) \(0.02629133951 - 0.05047233601i\)
\(L(1)\) \(\approx\) \(0.5523732176 + 0.2001588110i\)
\(L(1)\) \(\approx\) \(0.5523732176 + 0.2001588110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.0647 + 0.997i)T \)
3 \( 1 + (-0.985 - 0.168i)T \)
5 \( 1 + (-0.863 - 0.504i)T \)
7 \( 1 + (0.691 + 0.722i)T \)
11 \( 1 + (0.525 - 0.850i)T \)
13 \( 1 + (0.820 + 0.571i)T \)
17 \( 1 + (-0.327 + 0.944i)T \)
23 \( 1 + (-0.0348 - 0.999i)T \)
29 \( 1 + (-0.820 - 0.571i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (0.999 - 0.0299i)T \)
41 \( 1 + (-0.973 - 0.227i)T \)
43 \( 1 + (-0.797 - 0.603i)T \)
47 \( 1 + (-0.999 - 0.00997i)T \)
53 \( 1 + (0.676 - 0.736i)T \)
59 \( 1 + (-0.289 - 0.957i)T \)
61 \( 1 + (-0.438 - 0.898i)T \)
67 \( 1 + (0.270 + 0.962i)T \)
71 \( 1 + (-0.961 + 0.275i)T \)
73 \( 1 + (-0.661 + 0.749i)T \)
79 \( 1 + (-0.153 + 0.988i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.739 - 0.672i)T \)
97 \( 1 + (0.0946 - 0.995i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.50372941168896009713787531632, −17.99028977302430040436582365002, −17.76054535441693054911747708568, −16.74046026176647039079466641525, −16.21697146708134243322887292955, −15.11109284425456440674863424808, −14.79753420352472286414169797589, −13.67600218576232214960561077706, −13.1683429366065838780850662076, −12.170488349287661745123245929898, −11.74360391501453198451722235752, −11.135256814122763739718001167023, −10.7159613139325750718096083569, −10.01066483264696616868348808741, −9.25643186791254236199516523896, −8.28719806044125244485706479472, −7.4275496211154012375755999546, −7.002398914049526871961872916572, −5.86248640505655216852470858896, −4.87846053161444022261662894274, −4.48121471476062011378670665539, −3.694605486869451427252141180134, −3.08327002166274393707137587866, −1.6587059693464419281188216735, −1.14982498639064040071962405646, 0.02447503857670081443888941138, 1.05329773886748013825391847053, 1.880193300540577715566464179700, 3.63876457385203745763365913847, 4.186449549133379470038874107615, 4.85885062802648086610004019132, 5.677189052488059647731230777310, 6.19274798126998859789096449486, 6.85509810265372123432268182800, 7.83119167241874863103077738991, 8.420023476803210439874479200015, 8.836571317683004047562252358, 9.80858692553441583477978450381, 10.89224630162065702806719439067, 11.45366402126822850758275312924, 11.952576297628089115674096595096, 12.95752558006151614322166616143, 13.26064898608089511972108227318, 14.42190348006402481932079841923, 14.99597782053029382679676156594, 15.71138950473267928216706504812, 16.22946820242889001785706289959, 16.94889514756697598200985235420, 17.19107692124518972237741287015, 18.33406876181549749268132350204

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