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Properties

Label 2-2001-2001.275-c0-0-5
Degree $2$
Conductor $2001$
Sign $0.982 - 0.186i$
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 + (0.826 + 0.563i)3-s + (1.54 − 0.353i)4-s + (1.42 + 0.751i)6-s + (0.895 − 0.313i)8-s + (0.365 + 0.930i)9-s + (1.47 + 0.580i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (0.751 + 1.42i)18-s + (−0.781 + 0.623i)23-s + (0.916 + 0.245i)24-s + (−0.222 − 0.974i)25-s + (−1.69 − 2.69i)26-s + (−0.222 + 0.974i)27-s + ⋯
L(s)  = 1  + (1.59 − 0.180i)2-s + (0.826 + 0.563i)3-s + (1.54 − 0.353i)4-s + (1.42 + 0.751i)6-s + (0.895 − 0.313i)8-s + (0.365 + 0.930i)9-s + (1.47 + 0.580i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (0.751 + 1.42i)18-s + (−0.781 + 0.623i)23-s + (0.916 + 0.245i)24-s + (−0.222 − 0.974i)25-s + (−1.69 − 2.69i)26-s + (−0.222 + 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.982 - 0.186i$
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: \((2,\ 2001,\ (\ :0),\ 0.982 - 0.186i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.377559775\)
\(L(\frac12)\) \(\approx\) \(3.377559775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.826 - 0.563i)T \)
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

−9.570097971342106659099214956859, −8.398803192390780379241686468614, −7.85334614349850490859863928788, −6.86141089096520237293529419350, −5.84869188432178021488945014757, −5.05898057771749055231464003227, −4.55119041382554877547721960428, −3.41851288002817528729684072225, −3.01991910962949912987937214663, −2.01617376790695671350488797491, 1.92066128757736793348212094665, 2.57366555934229061934962893522, 3.71686559291027251610509331533, 4.27579188593782308516291979014, 5.16373291780663891848085222097, 6.28054794999237695542188373651, 6.77260156282408547972790706144, 7.45094989304151046205418262877, 8.382637612974282301333498054156, 9.316812595043610528883721380991

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