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Properties

Label 2-2001-2001.1172-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.981 - 0.189i$
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 − i)2-s + i·3-s − i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s − i·27-s − i·29-s + (−1 − i)31-s + (1 − i)32-s + ⋯
L(s)  = 1  + (1 − i)2-s + i·3-s − i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s − i·27-s − i·29-s + (−1 − i)31-s + (1 − i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.933768823\)
\(L(\frac12)\) \(\approx\) \(1.933768823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 + iT \)
good2 \( 1 + (-1 + i)T - iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - iT^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.448714402438412153078856361613, −9.013583199878180017521722825469, −7.909930291327359386067163393486, −6.82301674379719791647076645185, −5.77487939191565230315492397537, −5.06090099359527545749738494172, −4.14873997327109448197734214665, −3.85620080643315639556510783474, −2.71534346064884293610064227614, −1.80863970995744292176853033784, 1.11946198236062556720060919945, 2.76570478491314705304799496869, 3.48871735581238695571082916185, 4.84606035679535512913514622259, 5.46915395071510417297680507734, 6.08930402049363758613191143349, 7.01173845027374617510831424490, 7.38519636203980971489244978722, 8.311054210908338613961222226780, 8.818386907003194553262343213001

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