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Properties

Label 2-2001-2001.1448-c0-0-2
Degree $2$
Conductor $2001$
Sign $0.805 - 0.592i$
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 + (0.781 + 0.623i)3-s + (−0.926 − 0.211i)4-s + (−0.158 − 0.158i)6-s + (0.412 + 0.144i)8-s + (0.222 + 0.974i)9-s + (−0.592 − 0.742i)12-s + (0.541 − 1.12i)13-s + (0.767 + 0.369i)16-s + (−0.0250 − 0.222i)18-s + (0.781 + 0.623i)23-s + (0.232 + 0.369i)24-s + (−0.222 + 0.974i)25-s + (−0.148 + 0.236i)26-s + (−0.433 + 0.900i)27-s + ⋯
L(s)  = 1  + (−0.222 − 0.0250i)2-s + (0.781 + 0.623i)3-s + (−0.926 − 0.211i)4-s + (−0.158 − 0.158i)6-s + (0.412 + 0.144i)8-s + (0.222 + 0.974i)9-s + (−0.592 − 0.742i)12-s + (0.541 − 1.12i)13-s + (0.767 + 0.369i)16-s + (−0.0250 − 0.222i)18-s + (0.781 + 0.623i)23-s + (0.232 + 0.369i)24-s + (−0.222 + 0.974i)25-s + (−0.148 + 0.236i)26-s + (−0.433 + 0.900i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.805 - 0.592i$
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: \((2,\ 2001,\ (\ :0),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149143085\)
\(L(\frac12)\) \(\approx\) \(1.149143085\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.781 - 0.623i)T \)
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

−9.419793751861336711981719706396, −8.765316433327222234366156878150, −7.991354584225347152279755980577, −7.53125185520391169211932611928, −6.06062776537738905821480871039, −5.25774154126959543653108378247, −4.50244551083679416496164384324, −3.62152374103860928846360416954, −2.81856833613691042997145679893, −1.28022904199962708516313214069, 1.05559255762933578757781712889, 2.32180074601726344856533534853, 3.46443150750175638704438592816, 4.22043709217125762580017042173, 5.10545759767552508776844115708, 6.43840925313676481590651129314, 6.93355952090672692024948494010, 7.956459620547753135648399131319, 8.609922920377956105440096908285, 8.957768278728986689007762566344

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