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Properties

Label 1-2004-2004.1271-r1-0-0
Degree $1$
Conductor $2004$
Sign $0.922 + 0.387i$
Analytic cond. $215.359$
Root an. cond. $215.359$
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  + (−0.700 − 0.713i)5-s + (−0.898 + 0.438i)7-s + (−0.965 − 0.261i)11-s + (0.982 + 0.188i)13-s + (0.280 + 0.959i)17-s + (0.521 − 0.853i)19-s + (−0.988 − 0.150i)23-s + (−0.0189 + 0.999i)25-s + (−0.988 + 0.150i)29-s + (0.752 − 0.658i)31-s + (0.942 + 0.334i)35-s + (−0.206 − 0.978i)37-s + (0.132 − 0.991i)41-s + (−0.999 − 0.0378i)43-s + (−0.316 − 0.948i)47-s + ⋯
L(s)  = 1  + (−0.700 − 0.713i)5-s + (−0.898 + 0.438i)7-s + (−0.965 − 0.261i)11-s + (0.982 + 0.188i)13-s + (0.280 + 0.959i)17-s + (0.521 − 0.853i)19-s + (−0.988 − 0.150i)23-s + (−0.0189 + 0.999i)25-s + (−0.988 + 0.150i)29-s + (0.752 − 0.658i)31-s + (0.942 + 0.334i)35-s + (−0.206 − 0.978i)37-s + (0.132 − 0.991i)41-s + (−0.999 − 0.0378i)43-s + (−0.316 − 0.948i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $0.922 + 0.387i$
Analytic conductor: \(215.359\)
Root analytic conductor: \(215.359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2004} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2004,\ (1:\ ),\ 0.922 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7252374413 + 0.1460934376i\)
\(L(\frac12)\) \(\approx\) \(0.7252374413 + 0.1460934376i\)
\(L(1)\) \(\approx\) \(0.7151750479 - 0.06765401337i\)
\(L(1)\) \(\approx\) \(0.7151750479 - 0.06765401337i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.700 - 0.713i)T \)
7 \( 1 + (-0.898 + 0.438i)T \)
11 \( 1 + (-0.965 - 0.261i)T \)
13 \( 1 + (0.982 + 0.188i)T \)
17 \( 1 + (0.280 + 0.959i)T \)
19 \( 1 + (0.521 - 0.853i)T \)
23 \( 1 + (-0.988 - 0.150i)T \)
29 \( 1 + (-0.988 + 0.150i)T \)
31 \( 1 + (0.752 - 0.658i)T \)
37 \( 1 + (-0.206 - 0.978i)T \)
41 \( 1 + (0.132 - 0.991i)T \)
43 \( 1 + (-0.999 - 0.0378i)T \)
47 \( 1 + (-0.316 - 0.948i)T \)
53 \( 1 + (-0.243 + 0.969i)T \)
59 \( 1 + (-0.280 + 0.959i)T \)
61 \( 1 + (-0.584 + 0.811i)T \)
67 \( 1 + (-0.700 + 0.713i)T \)
71 \( 1 + (-0.351 - 0.936i)T \)
73 \( 1 + (-0.0567 + 0.998i)T \)
79 \( 1 + (-0.169 - 0.985i)T \)
83 \( 1 + (-0.929 + 0.369i)T \)
89 \( 1 + (0.993 - 0.113i)T \)
97 \( 1 + (-0.752 - 0.658i)T \)
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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

−19.78345903970652159400727751837, −18.73652578380210136419760937180, −18.53048346944323806043666802750, −17.73688409922064872190140830455, −16.545794795704375451677557467428, −15.96086542351673975404069156513, −15.59683418501172912656784555797, −14.596309299300257651385888042063, −13.78667033357681574615292951402, −13.210985230550107028275352351889, −12.28868154659729263125192352758, −11.5790122977124052664562625271, −10.76270131710405753953151662123, −10.069356216618896240056729435392, −9.51270425302665964975704594735, −8.08376012761098761599807336930, −7.8378663359911771193042351388, −6.81571607184017866910207963431, −6.217150234164241853933274745659, −5.21631309317862817223947252101, −4.16855475940139546266615058394, −3.293004803517820193246548849454, −2.91527417035143521134277878581, −1.51801028032332637152567380046, −0.26582899638838615854279729199, 0.45470785764017920761458996380, 1.635930033402385239023074495899, 2.77926715118219444890338088781, 3.640395114363285557773663096604, 4.29923613907611436259690373365, 5.51667076026147001778875265553, 5.90248099245784397729607639647, 7.03761656491366709008509949528, 7.88253175872832305032816111593, 8.608040012512204227781908018397, 9.1854195369292849442214253825, 10.1705783370112967425152888138, 10.92767977880857703764476580658, 11.820528839930561120130756913956, 12.4309116854281437394596138678, 13.275624886481610490727843322505, 13.57960499140889549705836379814, 15.00502573597883496737420035398, 15.53702740295215586502850031654, 16.17171422856869041093102829919, 16.61747471659288630839971971103, 17.68084390223756570254560626472, 18.54952881722898204067327504024, 19.046660942984327198964025572672, 19.828195090359058262707622478177

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