Properties

Label 2-2001-2001.1172-c0-0-5
Degree $2$
Conductor $2001$
Sign $-0.755 + 0.654i$
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  + (1.36 − 1.36i)2-s + (−0.5 + 0.866i)3-s − 2.73i·4-s + (0.499 + 1.86i)6-s + (−2.36 − 2.36i)8-s + (−0.499 − 0.866i)9-s + (2.36 + 1.36i)12-s i·13-s − 3.73·16-s + (−1.86 − 0.5i)18-s i·23-s + (3.23 − 0.866i)24-s + 25-s + (−1.36 − 1.36i)26-s + 0.999·27-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (−0.5 + 0.866i)3-s − 2.73i·4-s + (0.499 + 1.86i)6-s + (−2.36 − 2.36i)8-s + (−0.499 − 0.866i)9-s + (2.36 + 1.36i)12-s i·13-s − 3.73·16-s + (−1.86 − 0.5i)18-s i·23-s + (3.23 − 0.866i)24-s + 25-s + (−1.36 − 1.36i)26-s + 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.755 + 0.654i$
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: \((2,\ 2001,\ (\ :0),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.804499159\)
\(L(\frac12)\) \(\approx\) \(1.804499159\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.435702103098115619321157900280, −8.718497816846362269557616129836, −7.18561845736582172009699467884, −5.90195641384237950891184622507, −5.75929918737616251336155296170, −4.65871248255268265131606166992, −4.20239051079156926265511367031, −3.20905929515802293840864512172, −2.52186702690588701174707878200, −0.906663652201849071416924115675, 2.02217805169350827756242471509, 3.24115687234898642842969228624, 4.20312680401528447391217630653, 5.14137690229698200961062988902, 5.66790195661538608611410497912, 6.50566644860060250791129497869, 7.14585695258219659786175950834, 7.53445030012669371307274076689, 8.555156390291384607004176566470, 9.146594178449210740008320538972

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