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Properties

Label 2-2001-2001.137-c0-0-2
Degree $2$
Conductor $2001$
Sign $0.962 - 0.272i$
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  + (0.623 + 0.218i)2-s + (−0.433 + 0.900i)3-s + (−0.440 − 0.351i)4-s + (−0.467 + 0.467i)6-s + (−0.549 − 0.874i)8-s + (−0.623 − 0.781i)9-s + (0.507 − 0.244i)12-s + (1.75 − 0.400i)13-s + (−0.0263 − 0.115i)16-s + (−0.218 − 0.623i)18-s + (−0.433 + 0.900i)23-s + (1.02 − 0.115i)24-s + (0.623 − 0.781i)25-s + (1.18 + 0.133i)26-s + (0.974 − 0.222i)27-s + ⋯
L(s)  = 1  + (0.623 + 0.218i)2-s + (−0.433 + 0.900i)3-s + (−0.440 − 0.351i)4-s + (−0.467 + 0.467i)6-s + (−0.549 − 0.874i)8-s + (−0.623 − 0.781i)9-s + (0.507 − 0.244i)12-s + (1.75 − 0.400i)13-s + (−0.0263 − 0.115i)16-s + (−0.218 − 0.623i)18-s + (−0.433 + 0.900i)23-s + (1.02 − 0.115i)24-s + (0.623 − 0.781i)25-s + (1.18 + 0.133i)26-s + (0.974 − 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226739208\)
\(L(\frac12)\) \(\approx\) \(1.226739208\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.433 - 0.900i)T \)
23 \( 1 + (0.433 - 0.900i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
good2 \( 1 + (-0.623 - 0.218i)T + (0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.433 + 0.900i)T^{2} \)
13 \( 1 + (-1.75 + 0.400i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.974 - 0.222i)T^{2} \)
31 \( 1 + (-0.467 + 1.33i)T + (-0.781 - 0.623i)T^{2} \)
37 \( 1 + (-0.433 + 0.900i)T^{2} \)
41 \( 1 + (-1.19 + 1.19i)T - iT^{2} \)
43 \( 1 + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (-0.189 - 0.119i)T + (0.433 + 0.900i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 - 1.24iT - T^{2} \)
61 \( 1 + (-0.974 - 0.222i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.193 + 0.846i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.0739 + 0.211i)T + (-0.781 + 0.623i)T^{2} \)
79 \( 1 + (0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (-0.974 + 0.222i)T^{2} \)
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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

−9.297906404115256869179441245634, −8.861998914212730967873597602181, −7.927996332153613938930264579194, −6.57345229586693691346088768678, −5.96235092735779170368530048084, −5.45963438312379762172017224710, −4.41119120010882821349459674402, −3.91748638090881408048105335966, −3.00070842589473426903744912856, −0.990033349502577949822383483533, 1.21559059870066960333625837851, 2.53677183602777746486504616259, 3.48783870329856867173435881062, 4.46015617265511729932911200862, 5.27874241710077933171054471342, 6.19517488205074798333029889048, 6.69243689351507808966052872906, 7.87434394741036059009435495464, 8.464282101476009313973701699806, 9.000835305856390752837972334329

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