↑

Properties

Label 2-2001-2001.896-c0-0-1
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.791 − 0.497i)2-s + (0.294 + 0.955i)3-s + (−0.0549 − 0.114i)4-s + (0.241 − 0.902i)6-s + (−0.117 + 1.04i)8-s + (−0.826 + 0.563i)9-s + (0.0928 − 0.0861i)12-s + (1.49 + 1.19i)13-s + (0.534 − 0.670i)16-s + (0.933 − 0.0349i)18-s + (−0.974 − 0.222i)23-s + (−1.03 + 0.195i)24-s + (−0.900 + 0.433i)25-s + (−0.589 − 1.68i)26-s + (−0.781 − 0.623i)27-s + ⋯
L(s)  = 1  + (−0.791 − 0.497i)2-s + (0.294 + 0.955i)3-s + (−0.0549 − 0.114i)4-s + (0.241 − 0.902i)6-s + (−0.117 + 1.04i)8-s + (−0.826 + 0.563i)9-s + (0.0928 − 0.0861i)12-s + (1.49 + 1.19i)13-s + (0.534 − 0.670i)16-s + (0.933 − 0.0349i)18-s + (−0.974 − 0.222i)23-s + (−1.03 + 0.195i)24-s + (−0.900 + 0.433i)25-s + (−0.589 − 1.68i)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: $\chi_{2001} (896, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6911856985\)
\(L(\frac12)\) \(\approx\) \(0.6911856985\)
\(L(1)\) 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 \( 1 + (0.791 + 0.497i)T + (0.433 + 0.900i)T^{2} \)
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 \( 1 + (0.275 - 0.438i)T + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (-0.974 - 0.222i)T^{2} \)
41 \( 1 + (-0.839 - 0.839i)T + iT^{2} \)
43 \( 1 + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (-1.50 + 0.169i)T + (0.974 - 0.222i)T^{2} \)
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 \( 1 + (0.197 + 0.314i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (0.781 + 0.623i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.310567156521937844492402400126, −9.063246650369097066928231226737, −8.331316048491240779991947413987, −7.49013847134762350609913475731, −6.07670952517947971239839223753, −5.61045998024045755075919942097, −4.36253523903072033150032012545, −3.80432832133131014894677937994, −2.52276377144678057892763898463, −1.51276755747322289457651553378, 0.65741543915153506063655878348, 1.99999346097072798382739684290, 3.35429208082305341595727165331, 3.96368028440346003863587475240, 5.77218529097816136921449925162, 6.05832954599954500540705114885, 7.16459931266962984936429679388, 7.74254540302534817657646723373, 8.296604569007677497579988377725, 8.894397376913974889594845612418

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