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Properties

Label 2-2001-2001.482-c0-0-4
Degree $2$
Conductor $2001$
Sign $0.962 + 0.272i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
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  + (1.23 − 0.430i)2-s + (0.997 + 0.0747i)3-s + (0.548 − 0.437i)4-s + (1.26 − 0.337i)6-s + (−0.206 + 0.329i)8-s + (0.988 + 0.149i)9-s + (0.579 − 0.395i)12-s + (−0.145 − 0.0332i)13-s + (−0.269 + 1.17i)16-s + (1.28 − 0.242i)18-s + (−0.433 − 0.900i)23-s + (−0.230 + 0.312i)24-s + (0.623 + 0.781i)25-s + (−0.193 + 0.0218i)26-s + (0.974 + 0.222i)27-s + ⋯
L(s)  = 1  + (1.23 − 0.430i)2-s + (0.997 + 0.0747i)3-s + (0.548 − 0.437i)4-s + (1.26 − 0.337i)6-s + (−0.206 + 0.329i)8-s + (0.988 + 0.149i)9-s + (0.579 − 0.395i)12-s + (−0.145 − 0.0332i)13-s + (−0.269 + 1.17i)16-s + (1.28 − 0.242i)18-s + (−0.433 − 0.900i)23-s + (−0.230 + 0.312i)24-s + (0.623 + 0.781i)25-s + (−0.193 + 0.0218i)26-s + (0.974 + 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.860047042\)
\(L(\frac12)\) \(\approx\) \(2.860047042\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.997 - 0.0747i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (0.930 + 0.365i)T \)
good2 \( 1 + (-1.23 + 0.430i)T + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.433 - 0.900i)T^{2} \)
13 \( 1 + (0.145 + 0.0332i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.974 + 0.222i)T^{2} \)
31 \( 1 + (0.638 + 1.82i)T + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (-0.0528 - 0.0528i)T + iT^{2} \)
43 \( 1 + (-0.781 - 0.623i)T^{2} \)
47 \( 1 + (1.55 - 0.975i)T + (0.433 - 0.900i)T^{2} \)
53 \( 1 + (0.623 + 0.781i)T^{2} \)
59 \( 1 + 1.24iT - T^{2} \)
61 \( 1 + (-0.974 + 0.222i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.250 - 1.09i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.531 - 1.51i)T + (-0.781 - 0.623i)T^{2} \)
79 \( 1 + (0.433 + 0.900i)T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.781 + 0.623i)T^{2} \)
97 \( 1 + (-0.974 - 0.222i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.358976496039190604821025802371, −8.493956709784587564398986571599, −7.84381153009720210691645935315, −6.89457137675734726230479079772, −5.93481860292203255969959580227, −5.01724888430076301095408770973, −4.21172041634797126770108589386, −3.55976508050854091353391775293, −2.66936380428746742247037722407, −1.86070081484915904155973427588, 1.68641105878167340453924294066, 2.99680803331520131282515602542, 3.57250708186717431058959886651, 4.50142429364711200693939310545, 5.20790759415756223285761063243, 6.19360820413855727697365459458, 7.00279711966458062147183987377, 7.58141320388723007632179119121, 8.583716221969390615497590646996, 9.260974523302931432286447719117

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