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Properties

Label 2-2001-2001.1034-c0-0-2
Degree $2$
Conductor $2001$
Sign $0.310 + 0.950i$
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  + (−0.900 + 0.566i)2-s + (−0.974 + 0.222i)3-s + (0.0573 − 0.119i)4-s + (0.752 − 0.752i)6-s + (−0.103 − 0.917i)8-s + (0.900 − 0.433i)9-s + (−0.0294 + 0.128i)12-s + (−0.347 + 0.277i)13-s + (0.695 + 0.871i)16-s + (−0.566 + 0.900i)18-s + (−0.974 + 0.222i)23-s + (0.304 + 0.871i)24-s + (−0.900 − 0.433i)25-s + (0.156 − 0.446i)26-s + (−0.781 + 0.623i)27-s + ⋯
L(s)  = 1  + (−0.900 + 0.566i)2-s + (−0.974 + 0.222i)3-s + (0.0573 − 0.119i)4-s + (0.752 − 0.752i)6-s + (−0.103 − 0.917i)8-s + (0.900 − 0.433i)9-s + (−0.0294 + 0.128i)12-s + (−0.347 + 0.277i)13-s + (0.695 + 0.871i)16-s + (−0.566 + 0.900i)18-s + (−0.974 + 0.222i)23-s + (0.304 + 0.871i)24-s + (−0.900 − 0.433i)25-s + (0.156 − 0.446i)26-s + (−0.781 + 0.623i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.310 + 0.950i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1850444355\)
\(L(\frac12)\) \(\approx\) \(0.1850444355\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 - 0.222i)T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (-0.433 + 0.900i)T \)
good2 \( 1 + (0.900 - 0.566i)T + (0.433 - 0.900i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (0.347 - 0.277i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.781 + 0.623i)T^{2} \)
31 \( 1 + (0.752 + 1.19i)T + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
43 \( 1 + (-0.433 - 0.900i)T^{2} \)
47 \( 1 + (1.87 + 0.211i)T + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + 1.80iT - T^{2} \)
61 \( 1 + (0.781 + 0.623i)T^{2} \)
67 \( 1 + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.00 + 1.59i)T + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (0.974 - 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.781 - 0.623i)T^{2} \)
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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

−9.422160446969773351244792429755, −8.173854645770367972184099694790, −7.81520241778060449568628691658, −6.72411829057623208534529407411, −6.33361152742437892994057206926, −5.35115423489637033061646198353, −4.34366622433567817073939437616, −3.58453758321431328345111682418, −1.85172201752438169174746694581, −0.21802360695063940453925465212, 1.30442746908178771324491254731, 2.21900001978340163416269799817, 3.62299948864500400560486390539, 4.92111617922776630695952051776, 5.43910856918006970517931310594, 6.36533140428948270688110398518, 7.27051968912570809929529643631, 8.028531225836295607165431069438, 8.864071199871556643652299464137, 9.680983123155682177205427780249

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