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Properties

Label 2-2001-2001.827-c0-0-2
Degree $2$
Conductor $2001$
Sign $0.638 + 0.769i$
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  + (−0.132 − 1.18i)2-s + (0.826 + 0.563i)3-s + (−0.400 + 0.0913i)4-s + (0.554 − 1.05i)6-s + (−0.231 − 0.660i)8-s + (0.365 + 0.930i)9-s + (−0.382 − 0.149i)12-s + (0.858 + 1.78i)13-s + (−1.11 + 0.538i)16-s + (1.05 − 0.554i)18-s + (0.781 − 0.623i)23-s + (0.181 − 0.676i)24-s + (−0.222 − 0.974i)25-s + (1.98 − 1.24i)26-s + (−0.222 + 0.974i)27-s + ⋯
L(s)  = 1  + (−0.132 − 1.18i)2-s + (0.826 + 0.563i)3-s + (−0.400 + 0.0913i)4-s + (0.554 − 1.05i)6-s + (−0.231 − 0.660i)8-s + (0.365 + 0.930i)9-s + (−0.382 − 0.149i)12-s + (0.858 + 1.78i)13-s + (−1.11 + 0.538i)16-s + (1.05 − 0.554i)18-s + (0.781 − 0.623i)23-s + (0.181 − 0.676i)24-s + (−0.222 − 0.974i)25-s + (1.98 − 1.24i)26-s + (−0.222 + 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565904831\)
\(L(\frac12)\) \(\approx\) \(1.565904831\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.826 - 0.563i)T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (0.680 + 0.733i)T \)
good2 \( 1 + (0.132 + 1.18i)T + (-0.974 + 0.222i)T^{2} \)
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 \( 1 + (-0.858 - 1.78i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (-1.91 + 0.216i)T + (0.974 - 0.222i)T^{2} \)
37 \( 1 + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (-1.07 - 1.07i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (1.88 + 0.660i)T + (0.781 + 0.623i)T^{2} \)
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 \( 1 + (1.67 - 0.807i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.928 + 0.104i)T + (0.974 + 0.222i)T^{2} \)
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

−9.442902002726966860091219997819, −8.720465395030061309517861688683, −8.051514659510078430891465653703, −6.82247057980896463990698616677, −6.23081333016297956126928682075, −4.58140035497579639885484725894, −4.19928396654507830486938644440, −3.19770531170553349062774707137, −2.38157493179378968927626519956, −1.47971851647448996515718058460, 1.33804740205544358664665139221, 2.81807781747903561206178543263, 3.42972706337209136142349957227, 4.87769154995322287613539403750, 5.80224175083444598718624886410, 6.36949187271353123499507612706, 7.38527414341400167396196405696, 7.71484491169740418774411740055, 8.486816099979431552545173194833, 9.011676313297336553096948117514

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