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Properties

Label 2-2001-2001.620-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.918 + 0.394i$
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  + (0.122 + 0.350i)2-s + (−0.733 − 0.680i)3-s + (0.673 − 0.537i)4-s + (0.148 − 0.340i)6-s + (0.586 + 0.368i)8-s + (0.0747 + 0.997i)9-s + (−0.859 − 0.0643i)12-s + (1.61 + 0.367i)13-s + (0.134 − 0.589i)16-s + (−0.340 + 0.148i)18-s + (−0.433 − 0.900i)23-s + (−0.179 − 0.668i)24-s + (0.623 + 0.781i)25-s + (0.0687 + 0.610i)26-s + (0.623 − 0.781i)27-s + ⋯
L(s)  = 1  + (0.122 + 0.350i)2-s + (−0.733 − 0.680i)3-s + (0.673 − 0.537i)4-s + (0.148 − 0.340i)6-s + (0.586 + 0.368i)8-s + (0.0747 + 0.997i)9-s + (−0.859 − 0.0643i)12-s + (1.61 + 0.367i)13-s + (0.134 − 0.589i)16-s + (−0.340 + 0.148i)18-s + (−0.433 − 0.900i)23-s + (−0.179 − 0.668i)24-s + (0.623 + 0.781i)25-s + (0.0687 + 0.610i)26-s + (0.623 − 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268518289\)
\(L(\frac12)\) \(\approx\) \(1.268518289\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.733 + 0.680i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (-0.149 - 0.988i)T \)
good2 \( 1 + (-0.122 - 0.350i)T + (-0.781 + 0.623i)T^{2} \)
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 \( 1 + (-1.61 - 0.367i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.974 - 0.222i)T^{2} \)
31 \( 1 + (1.82 - 0.638i)T + (0.781 - 0.623i)T^{2} \)
37 \( 1 + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (-0.660 + 0.660i)T - iT^{2} \)
43 \( 1 + (0.781 + 0.623i)T^{2} \)
47 \( 1 + (-0.631 - 1.00i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (-0.443 + 1.94i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.754 + 0.264i)T + (0.781 + 0.623i)T^{2} \)
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

−9.168999110812529625591603057567, −8.385846110472302715264756464034, −7.43395933431584256430793307641, −6.83421403194898408802116893695, −6.17988890134589581737059399784, −5.55428119908729858040754711410, −4.74387833026659074891553275011, −3.45479131631686313925908564750, −2.05480184107074485419237080698, −1.22837660014012224975507460414, 1.30561391527506769766683580055, 2.72840122731260673378784348655, 3.78740572587848711894349421088, 4.16795060323313123612581655997, 5.55753101300473537555538390525, 6.08261185634416217160549344032, 6.93443504168911231400811946041, 7.84541796305862054447849146429, 8.665764241666222553709583254404, 9.542646946136784128428528328572

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