Properties

Label 2-2001-2001.482-c0-0-2
Degree $2$
Conductor $2001$
Sign $0.962 + 0.272i$
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  + (−1.85 + 0.649i)2-s + (−0.563 + 0.826i)3-s + (2.23 − 1.78i)4-s + (0.508 − 1.89i)6-s + (−1.94 + 3.09i)8-s + (−0.365 − 0.930i)9-s + (0.213 + 2.85i)12-s + (−1.61 − 0.367i)13-s + (0.963 − 4.21i)16-s + (1.28 + 1.48i)18-s + (−0.433 − 0.900i)23-s + (−1.46 − 3.35i)24-s + (0.623 + 0.781i)25-s + (3.22 − 0.363i)26-s + (0.974 + 0.222i)27-s + ⋯
L(s)  = 1  + (−1.85 + 0.649i)2-s + (−0.563 + 0.826i)3-s + (2.23 − 1.78i)4-s + (0.508 − 1.89i)6-s + (−1.94 + 3.09i)8-s + (−0.365 − 0.930i)9-s + (0.213 + 2.85i)12-s + (−1.61 − 0.367i)13-s + (0.963 − 4.21i)16-s + (1.28 + 1.48i)18-s + (−0.433 − 0.900i)23-s + (−1.46 − 3.35i)24-s + (0.623 + 0.781i)25-s + (3.22 − 0.363i)26-s + (0.974 + 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2498336603\)
\(L(\frac12)\) \(\approx\) \(0.2498336603\)
\(L(1)\) 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 \( 1 + (1.85 - 0.649i)T + (0.781 - 0.623i)T^{2} \)
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 \( 1 + (-0.170 - 0.488i)T + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (1.25 + 1.25i)T + iT^{2} \)
43 \( 1 + (-0.781 - 0.623i)T^{2} \)
47 \( 1 + (-1.36 + 0.856i)T + (0.433 - 0.900i)T^{2} \)
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 \( 1 + (-0.605 + 1.73i)T + (-0.781 - 0.623i)T^{2} \)
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

−9.215697586127163838011985181811, −8.777580697515086929918285605548, −7.85575399236191834918477698544, −7.00837715374523270214627923901, −6.51559351554426543964815666052, −5.37829332895455559474108124359, −4.97262826710390408327322979448, −3.21004332606270796239548112556, −2.03210236789722471793847289339, −0.40369101676443683258716682441, 1.06956138671293046332303533005, 2.21456796991750412196687853643, 2.79694522295272909101835227788, 4.36709788836012994747240863836, 5.78663238018108176010697976743, 6.70140657113048216815694213298, 7.28247236900386709128664450806, 7.906627096452322282044691102187, 8.530179447733104383310799556240, 9.597634898284639584522041022352

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