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Properties

Label 2-2001-2001.1931-c0-0-5
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: $\chi_{2001} (1931, \cdot )$
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

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

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