Properties

Label 2-2001-2001.206-c0-0-3
Degree $2$
Conductor $2001$
Sign $-0.521 - 0.853i$
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  + (−0.900 + 1.43i)2-s + (0.974 + 0.222i)3-s + (−0.810 − 1.68i)4-s + (−1.19 + 1.19i)6-s + (1.46 + 0.164i)8-s + (0.900 + 0.433i)9-s + (−0.415 − 1.82i)12-s + (0.347 + 0.277i)13-s + (−0.387 + 0.485i)16-s + (−1.43 + 0.900i)18-s + (0.974 + 0.222i)23-s + (1.38 + 0.485i)24-s + (−0.900 + 0.433i)25-s + (−0.711 + 0.248i)26-s + (0.781 + 0.623i)27-s + ⋯
L(s)  = 1  + (−0.900 + 1.43i)2-s + (0.974 + 0.222i)3-s + (−0.810 − 1.68i)4-s + (−1.19 + 1.19i)6-s + (1.46 + 0.164i)8-s + (0.900 + 0.433i)9-s + (−0.415 − 1.82i)12-s + (0.347 + 0.277i)13-s + (−0.387 + 0.485i)16-s + (−1.43 + 0.900i)18-s + (0.974 + 0.222i)23-s + (1.38 + 0.485i)24-s + (−0.900 + 0.433i)25-s + (−0.711 + 0.248i)26-s + (0.781 + 0.623i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014022319\)
\(L(\frac12)\) \(\approx\) \(1.014022319\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.974 - 0.222i)T \)
23 \( 1 + (-0.974 - 0.222i)T \)
29 \( 1 + (0.433 + 0.900i)T \)
good2 \( 1 + (0.900 - 1.43i)T + (-0.433 - 0.900i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.974 + 0.222i)T^{2} \)
13 \( 1 + (-0.347 - 0.277i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.781 + 0.623i)T^{2} \)
31 \( 1 + (-1.19 - 0.752i)T + (0.433 + 0.900i)T^{2} \)
37 \( 1 + (0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.158 + 0.158i)T - iT^{2} \)
43 \( 1 + (0.433 - 0.900i)T^{2} \)
47 \( 1 + (-0.0739 - 0.656i)T + (-0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + 1.80iT - T^{2} \)
61 \( 1 + (-0.781 + 0.623i)T^{2} \)
67 \( 1 + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.559 - 0.351i)T + (0.433 - 0.900i)T^{2} \)
79 \( 1 + (-0.974 - 0.222i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.433 + 0.900i)T^{2} \)
97 \( 1 + (-0.781 - 0.623i)T^{2} \)
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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

−9.354375159977853069867328053516, −8.703747417080700604326694857743, −8.066174363673156350058764683868, −7.43906110709513718285693864166, −6.75098360553972472977521807623, −5.89503932554467345387337060847, −4.95577255689630978394102725256, −4.00386050507714591078603173112, −2.81714908532823150658520983667, −1.38548675979364395157753701400, 1.04591744591398606015232252720, 2.10026283813814153116905633521, 2.91094564209163515288697300508, 3.66536308297317511316081981009, 4.54465086309792356039472470332, 6.00738082805637352622339639953, 7.16824785216302396996083559641, 7.88139673977076624660903912216, 8.660342816148495825957294149486, 9.034482791200691377992584534995

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