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Properties

Label 2-2001-2001.1448-c0-0-1
Degree $2$
Conductor $2001$
Sign $-0.329 - 0.944i$
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  + (−1.82 − 0.205i)2-s + (0.0747 + 0.997i)3-s + (2.30 + 0.525i)4-s + (0.0685 − 1.83i)6-s + (−2.35 − 0.823i)8-s + (−0.988 + 0.149i)9-s + (−0.351 + 2.33i)12-s + (0.317 − 0.658i)13-s + (1.99 + 0.959i)16-s + (1.83 − 0.0685i)18-s + (−0.781 − 0.623i)23-s + (0.645 − 2.40i)24-s + (−0.222 + 0.974i)25-s + (−0.712 + 1.13i)26-s + (−0.222 − 0.974i)27-s + ⋯
L(s)  = 1  + (−1.82 − 0.205i)2-s + (0.0747 + 0.997i)3-s + (2.30 + 0.525i)4-s + (0.0685 − 1.83i)6-s + (−2.35 − 0.823i)8-s + (−0.988 + 0.149i)9-s + (−0.351 + 2.33i)12-s + (0.317 − 0.658i)13-s + (1.99 + 0.959i)16-s + (1.83 − 0.0685i)18-s + (−0.781 − 0.623i)23-s + (0.645 − 2.40i)24-s + (−0.222 + 0.974i)25-s + (−0.712 + 1.13i)26-s + (−0.222 − 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4049773576\)
\(L(\frac12)\) \(\approx\) \(0.4049773576\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (0.781 + 0.623i)T \)
29 \( 1 + (-0.294 - 0.955i)T \)
good2 \( 1 + (1.82 + 0.205i)T + (0.974 + 0.222i)T^{2} \)
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 \( 1 + (-0.317 + 0.658i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.433 + 0.900i)T^{2} \)
31 \( 1 + (0.216 - 1.91i)T + (-0.974 - 0.222i)T^{2} \)
37 \( 1 + (0.781 + 0.623i)T^{2} \)
41 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
43 \( 1 + (-0.974 + 0.222i)T^{2} \)
47 \( 1 + (-0.584 - 1.66i)T + (-0.781 + 0.623i)T^{2} \)
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 \( 1 + (0.268 + 0.129i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.00837 + 0.0743i)T + (-0.974 + 0.222i)T^{2} \)
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

−9.463363651812825403125324909826, −9.003805542940482517652783306906, −8.230603444815428217545675403472, −7.69711246208561049971270705276, −6.63995539877719582092083007278, −5.81833068311709813141475904699, −4.71377084984173116892604850396, −3.40304240840418718260915861266, −2.70585673477413269350367147354, −1.32291495589009466147394620293, 0.54848036853844730473019349729, 1.88205690466628513365784688854, 2.44829314163191406724526771796, 3.97412916633236516571457341200, 5.76846075532926637792861393009, 6.23347382357981269076510613006, 7.09959847354561783315851377374, 7.69510776160540458941528131670, 8.296241153703000780475872172151, 8.972174275315096478549266124025

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