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Properties

Label 2-2001-2001.275-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.805 + 0.592i$
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  + (−1.59 + 0.180i)2-s + (0.149 + 0.988i)3-s + (1.54 − 0.353i)4-s + (−0.416 − 1.55i)6-s + (−0.895 + 0.313i)8-s + (−0.955 + 0.294i)9-s + (0.580 + 1.47i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (1.47 − 0.643i)18-s + (0.781 − 0.623i)23-s + (−0.443 − 0.838i)24-s + (−0.222 − 0.974i)25-s + (1.69 + 2.69i)26-s + (−0.433 − 0.900i)27-s + ⋯
L(s)  = 1  + (−1.59 + 0.180i)2-s + (0.149 + 0.988i)3-s + (1.54 − 0.353i)4-s + (−0.416 − 1.55i)6-s + (−0.895 + 0.313i)8-s + (−0.955 + 0.294i)9-s + (0.580 + 1.47i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (1.47 − 0.643i)18-s + (0.781 − 0.623i)23-s + (−0.443 − 0.838i)24-s + (−0.222 − 0.974i)25-s + (1.69 + 2.69i)26-s + (−0.433 − 0.900i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3791931340\)
\(L(\frac12)\) \(\approx\) \(0.3791931340\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.149 - 0.988i)T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (0.680 + 0.733i)T \)
good2 \( 1 + (1.59 - 0.180i)T + (0.974 - 0.222i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.781 - 0.623i)T^{2} \)
13 \( 1 + (0.858 + 1.78i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.433 - 0.900i)T^{2} \)
31 \( 1 + (-0.0579 - 0.514i)T + (-0.974 + 0.222i)T^{2} \)
37 \( 1 + (0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.922 + 0.922i)T - iT^{2} \)
43 \( 1 + (-0.974 - 0.222i)T^{2} \)
47 \( 1 + (-0.0246 + 0.0705i)T + (-0.781 - 0.623i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + 0.445iT - T^{2} \)
61 \( 1 + (0.433 - 0.900i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (1.67 - 0.807i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.197 + 1.75i)T + (-0.974 - 0.222i)T^{2} \)
79 \( 1 + (-0.781 + 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.974 + 0.222i)T^{2} \)
97 \( 1 + (0.433 + 0.900i)T^{2} \)
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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

−9.317500993856453718514265040136, −8.555476775006910411435631316620, −7.976315031443506571811032036834, −7.33158945690752095081958659751, −6.22157426623057180155281846220, −5.34395744439267694143868670167, −4.42319794278350964229546064996, −3.13849797584546453276943521911, −2.28446777965556325854627972817, −0.46631641788245242496597764047, 1.34704343865213695524546033135, 2.04995994478406305479024245500, 3.08036491626361064034784785001, 4.55057758438707294230821504035, 5.79301009591320186822018727975, 6.86473025130129406926147897593, 7.21309014014431188038728810704, 7.86720748497686875081525666474, 8.794987040771166178497253950126, 9.312130691303281009317111499364

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