Properties

Label 1-2001-2001.344-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.899 - 0.437i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)8-s + (−0.623 − 0.781i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + 17-s + (−0.623 − 0.781i)19-s + (−0.222 − 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.623 + 0.781i)25-s + (−0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)8-s + (−0.623 − 0.781i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + 17-s + (−0.623 − 0.781i)19-s + (−0.222 − 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.623 + 0.781i)25-s + (−0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.899 - 0.437i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2214269714 + 0.9615316008i\)
\(L(\frac12)\) \(\approx\) \(-0.2214269714 + 0.9615316008i\)
\(L(1)\) \(\approx\) \(1.012975503 + 0.6272614973i\)
\(L(1)\) \(\approx\) \(1.012975503 + 0.6272614973i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 - T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.222 + 0.974i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.623 - 0.781i)T \)
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

−19.65667182766721323800296348893, −18.97319800858774845365637264083, −18.52339822859865865730143661547, −17.18130214591466168527650182791, −16.260909495952725405398520365232, −15.937376656489715107329930102582, −14.965921853168391567305812059998, −14.35339252256970780073074783451, −13.71956548532892091979041520070, −12.61922353066647096533472376047, −12.49468994933003142992629643009, −11.34524937804390117319238311486, −10.63659696782204796774071291058, −10.35011367323704504846009064602, −9.23424178911399885130713309891, −7.86692485767180947468764483084, −7.53705070783927425608388583948, −6.41942719080587142404298433072, −5.83907281535557434214276263177, −4.842825425250825374470063397782, −3.819424097342070980565168913174, −3.3855208657875125787890870151, −2.742383836899910396496987323371, −1.28443532492465665738518185013, −0.240860439900751317714303984951, 1.72905564402205764883874865421, 2.65432232562851369384632484279, 3.5323294023388293475983910013, 4.36724101323432506811363654348, 4.99147023995247876988764695217, 5.83698468003668396285187644091, 6.84649692285700412299285781567, 7.317227525822268795436707735420, 8.268873640436364552476890932494, 8.99204121849686319943291395451, 9.87056713815159273360995109103, 11.080486695088252967525761809249, 11.84326450590464649905944974695, 12.35492614400479359051406730404, 12.86432260585022425031020899947, 13.73833086890242158413982929699, 14.79751336463091741906077601594, 15.1944492152756724619711022436, 15.81968013124752098034725283695, 16.63601261578222500363849673725, 17.02990761473396958452881814891, 18.24711327470996622619604365654, 19.04130395154002909194098946042, 19.724935335733374458341264006714, 20.490678172770687606771579124558

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