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Properties

Label 2-2001-2001.896-c0-0-5
Degree $2$
Conductor $2001$
Sign $0.721 + 0.692i$
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 + 0.566i)2-s + (0.623 − 0.781i)3-s + (0.0573 + 0.119i)4-s + (1.00 − 0.351i)6-s + (0.103 − 0.917i)8-s + (−0.222 − 0.974i)9-s + (0.128 + 0.0294i)12-s + (−0.347 − 0.277i)13-s + (0.695 − 0.871i)16-s + (0.351 − 1.00i)18-s + (0.974 + 0.222i)23-s + (−0.652 − 0.652i)24-s + (−0.900 + 0.433i)25-s + (−0.156 − 0.446i)26-s + (−0.900 − 0.433i)27-s + ⋯
L(s)  = 1  + (0.900 + 0.566i)2-s + (0.623 − 0.781i)3-s + (0.0573 + 0.119i)4-s + (1.00 − 0.351i)6-s + (0.103 − 0.917i)8-s + (−0.222 − 0.974i)9-s + (0.128 + 0.0294i)12-s + (−0.347 − 0.277i)13-s + (0.695 − 0.871i)16-s + (0.351 − 1.00i)18-s + (0.974 + 0.222i)23-s + (−0.652 − 0.652i)24-s + (−0.900 + 0.433i)25-s + (−0.156 − 0.446i)26-s + (−0.900 − 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (896, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.721 + 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.110058098\)
\(L(\frac12)\) \(\approx\) \(2.110058098\)
\(L(1)\) 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 \( 1 + (-0.900 - 0.566i)T + (0.433 + 0.900i)T^{2} \)
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 \( 1 + (0.752 - 1.19i)T + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (-0.974 - 0.222i)T^{2} \)
41 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
43 \( 1 + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (-1.87 + 0.211i)T + (0.974 - 0.222i)T^{2} \)
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 \( 1 + (-1.00 - 1.59i)T + (-0.433 + 0.900i)T^{2} \)
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

−9.276953491959918336817724256920, −8.253059300704922948237721930277, −7.40235069412211143076959549578, −6.98113848039978281428231833684, −5.99403649681665138431307459286, −5.44794287897038173086485036631, −4.34444711095935255773382017481, −3.52189181872127924100766738947, −2.54032805458544618333110414543, −1.16694647723800286705243893062, 2.07233789720136915399957026876, 2.77208973828410978020221550142, 3.81516815669553219338853763892, 4.25005955194709638058286507824, 5.20241534865359307806896813453, 5.83619942490489855130852418951, 7.31070679019885303354655660689, 7.86881471650011262897755991577, 9.068014657020446075664432857786, 9.139467085364633200394300254832

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