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Properties

Label 2-2001-2001.206-c0-0-5
Degree $2$
Conductor $2001$
Sign $-0.521 - 0.853i$
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  + (0.940 − 1.49i)2-s + (−0.294 − 0.955i)3-s + (−0.922 − 1.91i)4-s + (−1.70 − 0.457i)6-s + (−1.97 − 0.223i)8-s + (−0.826 + 0.563i)9-s + (−1.55 + 1.44i)12-s + (−1.49 − 1.19i)13-s + (−0.870 + 1.09i)16-s + (0.0661 + 1.76i)18-s + (0.974 + 0.222i)23-s + (0.370 + 1.95i)24-s + (−0.900 + 0.433i)25-s + (−3.18 + 1.11i)26-s + (0.781 + 0.623i)27-s + ⋯
L(s)  = 1  + (0.940 − 1.49i)2-s + (−0.294 − 0.955i)3-s + (−0.922 − 1.91i)4-s + (−1.70 − 0.457i)6-s + (−1.97 − 0.223i)8-s + (−0.826 + 0.563i)9-s + (−1.55 + 1.44i)12-s + (−1.49 − 1.19i)13-s + (−0.870 + 1.09i)16-s + (0.0661 + 1.76i)18-s + (0.974 + 0.222i)23-s + (0.370 + 1.95i)24-s + (−0.900 + 0.433i)25-s + (−3.18 + 1.11i)26-s + (0.781 + 0.623i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274295332\)
\(L(\frac12)\) \(\approx\) \(1.274295332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.294 + 0.955i)T \)
23 \( 1 + (-0.974 - 0.222i)T \)
29 \( 1 + (-0.997 - 0.0747i)T \)
good2 \( 1 + (-0.940 + 1.49i)T + (-0.433 - 0.900i)T^{2} \)
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 \( 1 + (1.49 + 1.19i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.781 + 0.623i)T^{2} \)
31 \( 1 + (1.63 + 1.02i)T + (0.433 + 0.900i)T^{2} \)
37 \( 1 + (0.974 + 0.222i)T^{2} \)
41 \( 1 + (-1.13 + 1.13i)T - iT^{2} \)
43 \( 1 + (0.433 - 0.900i)T^{2} \)
47 \( 1 + (-0.146 - 1.29i)T + (-0.974 + 0.222i)T^{2} \)
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 \( 1 + (-0.848 + 1.06i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.66 + 1.04i)T + (0.433 - 0.900i)T^{2} \)
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

−9.200389751447975067400349029142, −7.84395489426776019215160200275, −7.34823030915175378877372113997, −6.10123158987148333867174875181, −5.37786230613224279681895826705, −4.81487952063188905024389235400, −3.57893458811757370920675778753, −2.68485149845397789082808845166, −2.00781232090479108012899740836, −0.68140438543061815252031011930, 2.60105092615514459905410929078, 3.77622980518830526980438694708, 4.44589020372508515528893450723, 5.09195192658708625878642189237, 5.71677452110442095613420402921, 6.75729660623488769572183637444, 7.12061764005104217614802553443, 8.194113789545654045290916818791, 8.964070658379146163613722135653, 9.607279155038357484789751008104

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