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Properties

Label 2-2001-2001.1448-c0-0-3
Degree $2$
Conductor $2001$
Sign $-0.653 + 0.757i$
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  + (0.222 + 0.0250i)2-s + (−0.900 − 0.433i)3-s + (−0.926 − 0.211i)4-s + (−0.189 − 0.119i)6-s + (−0.412 − 0.144i)8-s + (0.623 + 0.781i)9-s + (0.742 + 0.592i)12-s + (0.541 − 1.12i)13-s + (0.767 + 0.369i)16-s + (0.119 + 0.189i)18-s + (−0.781 − 0.623i)23-s + (0.308 + 0.308i)24-s + (−0.222 + 0.974i)25-s + (0.148 − 0.236i)26-s + (−0.222 − 0.974i)27-s + ⋯
L(s)  = 1  + (0.222 + 0.0250i)2-s + (−0.900 − 0.433i)3-s + (−0.926 − 0.211i)4-s + (−0.189 − 0.119i)6-s + (−0.412 − 0.144i)8-s + (0.623 + 0.781i)9-s + (0.742 + 0.592i)12-s + (0.541 − 1.12i)13-s + (0.767 + 0.369i)16-s + (0.119 + 0.189i)18-s + (−0.781 − 0.623i)23-s + (0.308 + 0.308i)24-s + (−0.222 + 0.974i)25-s + (0.148 − 0.236i)26-s + (−0.222 − 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.653 + 0.757i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.653 + 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4899105839\)
\(L(\frac12)\) \(\approx\) \(0.4899105839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.781 + 0.623i)T \)
29 \( 1 + (0.974 + 0.222i)T \)
good2 \( 1 + (-0.222 - 0.0250i)T + (0.974 + 0.222i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.781 + 0.623i)T^{2} \)
13 \( 1 + (-0.541 + 1.12i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.433 + 0.900i)T^{2} \)
31 \( 1 + (-0.158 + 1.40i)T + (-0.974 - 0.222i)T^{2} \)
37 \( 1 + (0.781 + 0.623i)T^{2} \)
41 \( 1 + (0.467 + 0.467i)T + iT^{2} \)
43 \( 1 + (-0.974 + 0.222i)T^{2} \)
47 \( 1 + (0.559 + 1.59i)T + (-0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + 0.445iT - T^{2} \)
61 \( 1 + (0.433 + 0.900i)T^{2} \)
67 \( 1 + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.189 + 1.68i)T + (-0.974 + 0.222i)T^{2} \)
79 \( 1 + (-0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.974 - 0.222i)T^{2} \)
97 \( 1 + (0.433 - 0.900i)T^{2} \)
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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

−9.104208759149889099177445691093, −8.126819094404102771457155288710, −7.61194125131598105230099269666, −6.43024190909149532517858182116, −5.77129134521056008086894647251, −5.21492863891865928401096444128, −4.27978037522654744552997780645, −3.39092220515726638316809842567, −1.80996401662873055210617973241, −0.40056080148540021042572560132, 1.48041989502667604295017630805, 3.25909593652186449835958884096, 4.12541968146370911309260514662, 4.66517958315841377095476573516, 5.58581337200602978523321023405, 6.26901119540187539596968982453, 7.14651234597918292583616740260, 8.233908929724533489052606534504, 8.962733938688124393741708140300, 9.679764245814859376223643708444

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