Properties

Label 2-2001-2001.206-c0-0-4
Degree $2$
Conductor $2001$
Sign $-0.995 + 0.0997i$
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 + (0.623 − 0.781i)3-s + (−0.810 − 1.68i)4-s + (−0.559 − 1.59i)6-s + (−1.46 − 0.164i)8-s + (−0.222 − 0.974i)9-s + (−1.82 − 0.415i)12-s + (0.347 + 0.277i)13-s + (−0.387 + 0.485i)16-s + (−1.59 − 0.559i)18-s + (−0.974 − 0.222i)23-s + (−1.03 + 1.03i)24-s + (−0.900 + 0.433i)25-s + (0.711 − 0.248i)26-s + (−0.900 − 0.433i)27-s + ⋯
L(s)  = 1  + (0.900 − 1.43i)2-s + (0.623 − 0.781i)3-s + (−0.810 − 1.68i)4-s + (−0.559 − 1.59i)6-s + (−1.46 − 0.164i)8-s + (−0.222 − 0.974i)9-s + (−1.82 − 0.415i)12-s + (0.347 + 0.277i)13-s + (−0.387 + 0.485i)16-s + (−1.59 − 0.559i)18-s + (−0.974 − 0.222i)23-s + (−1.03 + 1.03i)24-s + (−0.900 + 0.433i)25-s + (0.711 − 0.248i)26-s + (−0.900 − 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.995 + 0.0997i$
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: \((2,\ 2001,\ (\ :0),\ -0.995 + 0.0997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.141797210\)
\(L(\frac12)\) \(\approx\) \(2.141797210\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.104854719991092399194593604001, −8.342454853377084904571170299494, −7.42372256683071457671124364597, −6.43157222350187737731536063157, −5.65739466991993815742052384138, −4.56067290256980701126197138431, −3.74356672875853793723736063057, −2.97453377920416241530484285525, −2.09124909154696315457701405354, −1.20406104806879388650302671359, 2.33713761154925118500050241129, 3.54168703298228365615105393226, 4.15430059526751539138538093864, 4.89093088620405652667517717260, 5.78880222651920852442435500629, 6.35379435364425944430060581725, 7.45213977845549572814994248294, 8.130984573410924484093652306194, 8.488712764208218614405356308111, 9.650239061924131414170486370264

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