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Properties

Label 2-2001-2001.1172-c0-0-0
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  + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-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

\(L(\frac{1}{2})\) \(\approx\) \(1.084475138\)
\(L(\frac12)\) \(\approx\) \(1.084475138\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - iT^{2} \)
31 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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

−9.409876902396886075578402610684, −8.364215319625513677218455031018, −7.73204663749347350574215234803, −7.07821217227158131102089797499, −6.16787685037953061378513476204, −5.22513946429756488004515141268, −4.63429578908161776856387020092, −3.46677617296152623034344502172, −2.61322163691978724226645359645, −1.29856665633053463670349467816, 0.915586976546599552619892365333, 2.38730044178192699650682811334, 4.15565750339888851861985690707, 4.35922411813746182554232787230, 5.45284178150281076795511102209, 6.02092574292142631439586123884, 6.74931738114314809733842765572, 7.40134775538651311991797932163, 8.785602891370929991581599454193, 9.411785078926859909802694611019

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