Properties

Label 2-2001-2001.620-c0-0-5
Degree $2$
Conductor $2001$
Sign $-0.253 + 0.967i$
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  + (−0.122 − 0.350i)2-s + (0.563 − 0.826i)3-s + (0.673 − 0.537i)4-s + (−0.359 − 0.0962i)6-s + (−0.586 − 0.368i)8-s + (−0.365 − 0.930i)9-s + (−0.0643 − 0.859i)12-s + (1.61 + 0.367i)13-s + (0.134 − 0.589i)16-s + (−0.281 + 0.242i)18-s + (0.433 + 0.900i)23-s + (−0.634 + 0.276i)24-s + (0.623 + 0.781i)25-s + (−0.0687 − 0.610i)26-s + (−0.974 − 0.222i)27-s + ⋯
L(s)  = 1  + (−0.122 − 0.350i)2-s + (0.563 − 0.826i)3-s + (0.673 − 0.537i)4-s + (−0.359 − 0.0962i)6-s + (−0.586 − 0.368i)8-s + (−0.365 − 0.930i)9-s + (−0.0643 − 0.859i)12-s + (1.61 + 0.367i)13-s + (0.134 − 0.589i)16-s + (−0.281 + 0.242i)18-s + (0.433 + 0.900i)23-s + (−0.634 + 0.276i)24-s + (0.623 + 0.781i)25-s + (−0.0687 − 0.610i)26-s + (−0.974 − 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.253 + 0.967i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.253 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.568522378\)
\(L(\frac12)\) \(\approx\) \(1.568522378\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.563 + 0.826i)T \)
23 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 + (0.149 + 0.988i)T \)
good2 \( 1 + (0.122 + 0.350i)T + (-0.781 + 0.623i)T^{2} \)
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 \( 1 + (-1.61 - 0.367i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.974 - 0.222i)T^{2} \)
31 \( 1 + (1.82 - 0.638i)T + (0.781 - 0.623i)T^{2} \)
37 \( 1 + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (0.660 - 0.660i)T - iT^{2} \)
43 \( 1 + (0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.631 + 1.00i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (0.443 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.754 + 0.264i)T + (0.781 + 0.623i)T^{2} \)
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

−9.022760834183275995906745163276, −8.503908179704598716645229706957, −7.39509890971605313835595977606, −6.87589917294847579488011123227, −6.07024260998943330155283762417, −5.38469476780045361656433651421, −3.74882319212210966561247769035, −3.13500791407756157716118916220, −1.90931047569524651794644609896, −1.24253666255304944698498926373, 1.87695689182312941589480882495, 3.06869809704114481014248790347, 3.58377275895689628562002154963, 4.63690124164868306548319173062, 5.69986597122469064861999683863, 6.42759137449149729654532132907, 7.37454327767040735407883088949, 8.156604985971034291425257665309, 8.733819150351100092167918922407, 9.265728928815717303201003324057

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