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Properties

Label 2-2001-2001.68-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.411 + 0.911i$
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  + (0.0397 + 0.0633i)2-s + (−0.988 + 0.149i)3-s + (0.431 − 0.895i)4-s + (−0.0487 − 0.0566i)6-s + (0.148 − 0.0166i)8-s + (0.955 − 0.294i)9-s + (−0.293 + 0.950i)12-s + (1.14 − 0.914i)13-s + (−0.613 − 0.768i)16-s + (0.0566 + 0.0487i)18-s + (−0.974 + 0.222i)23-s + (−0.144 + 0.0386i)24-s + (−0.900 − 0.433i)25-s + (0.103 + 0.0362i)26-s + (−0.900 + 0.433i)27-s + ⋯
L(s)  = 1  + (0.0397 + 0.0633i)2-s + (−0.988 + 0.149i)3-s + (0.431 − 0.895i)4-s + (−0.0487 − 0.0566i)6-s + (0.148 − 0.0166i)8-s + (0.955 − 0.294i)9-s + (−0.293 + 0.950i)12-s + (1.14 − 0.914i)13-s + (−0.613 − 0.768i)16-s + (0.0566 + 0.0487i)18-s + (−0.974 + 0.222i)23-s + (−0.144 + 0.0386i)24-s + (−0.900 − 0.433i)25-s + (0.103 + 0.0362i)26-s + (−0.900 + 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9586012524\)
\(L(\frac12)\) \(\approx\) \(0.9586012524\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (-0.563 - 0.826i)T \)
good2 \( 1 + (-0.0397 - 0.0633i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (-1.14 + 0.914i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
31 \( 1 + (-0.438 + 0.275i)T + (0.433 - 0.900i)T^{2} \)
37 \( 1 + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (-1.29 - 1.29i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (-0.220 + 1.95i)T + (-0.974 - 0.222i)T^{2} \)
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 \( 1 + (0.367 + 0.460i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.10 + 0.694i)T + (0.433 + 0.900i)T^{2} \)
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

−9.475048683129884212659201937013, −8.375123478983710933656968222561, −7.50293956932629048832323380664, −6.52343779547080411185740828644, −6.01367924179728442171234863817, −5.41788823718981442314860956059, −4.51049996689049276617589088787, −3.48212132487837464249331789653, −1.99534114763329552592234682278, −0.838728400105434187613977298708, 1.47227113317181140138268875290, 2.60432312411150854515388345945, 4.05079524074088330093304810425, 4.29264857376533661677114065145, 5.83870171313203494778864013206, 6.20619925107100464151468137784, 7.15201768241455304754869173313, 7.75678728461226433252116746567, 8.630125552761424854104545942814, 9.487383798008346775084306214006

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