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Properties

Label 2-2001-2001.620-c0-0-1
Degree $2$
Conductor $2001$
Sign $-0.253 + 0.967i$
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  + (−0.500 − 1.43i)2-s + (−0.997 − 0.0747i)3-s + (−1.01 + 0.809i)4-s + (0.392 + 1.46i)6-s + (0.382 + 0.240i)8-s + (0.988 + 0.149i)9-s + (1.07 − 0.731i)12-s + (0.145 + 0.0332i)13-s + (−0.136 + 0.597i)16-s + (−0.281 − 1.48i)18-s + (0.433 + 0.900i)23-s + (−0.363 − 0.268i)24-s + (0.623 + 0.781i)25-s + (−0.0253 − 0.225i)26-s + (−0.974 − 0.222i)27-s + ⋯
L(s)  = 1  + (−0.500 − 1.43i)2-s + (−0.997 − 0.0747i)3-s + (−1.01 + 0.809i)4-s + (0.392 + 1.46i)6-s + (0.382 + 0.240i)8-s + (0.988 + 0.149i)9-s + (1.07 − 0.731i)12-s + (0.145 + 0.0332i)13-s + (−0.136 + 0.597i)16-s + (−0.281 − 1.48i)18-s + (0.433 + 0.900i)23-s + (−0.363 − 0.268i)24-s + (0.623 + 0.781i)25-s + (−0.0253 − 0.225i)26-s + (−0.974 − 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.253 + 0.967i$
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: \((2,\ 2001,\ (\ :0),\ -0.253 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6109857838\)
\(L(\frac12)\) \(\approx\) \(0.6109857838\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 + 0.0747i)T \)
23 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 + (-0.930 - 0.365i)T \)
good2 \( 1 + (0.500 + 1.43i)T + (-0.781 + 0.623i)T^{2} \)
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 \( 1 + (-0.145 - 0.0332i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.974 - 0.222i)T^{2} \)
31 \( 1 + (-0.488 + 0.170i)T + (0.781 - 0.623i)T^{2} \)
37 \( 1 + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
43 \( 1 + (0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.425 + 0.677i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (-0.250 + 1.09i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (1.12 + 0.392i)T + (0.781 + 0.623i)T^{2} \)
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

−9.334442387278584863220475385570, −8.722614360833299153918362522642, −7.59610972693165868332104068767, −6.78024710421630693625529300767, −5.85768649278835812571517891725, −4.95421362492855160483549968583, −4.00822650614376073283273630875, −3.08405241351424276721125114070, −1.88505395620652904236627203866, −0.886596151832261877986956171824, 0.913867440450584866443718843494, 2.79216919974328299795538248257, 4.43778788675996687178755143427, 4.89762591024071034136375864646, 6.00532044038951468258889643957, 6.36172954292009991401547768754, 7.07652733869140293297933919008, 7.921461759851136742074745692927, 8.572029261017803371127623072587, 9.476907116582544097729363949013

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