↑

Properties

Label 2-2001-2001.482-c0-0-0
Degree $2$
Conductor $2001$
Sign $-0.582 - 0.813i$
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  + (−0.623 + 0.218i)2-s + (−0.222 + 0.974i)3-s + (−0.440 + 0.351i)4-s + (−0.0739 − 0.656i)6-s + (0.549 − 0.874i)8-s + (−0.900 − 0.433i)9-s + (−0.244 − 0.507i)12-s + (1.75 + 0.400i)13-s + (−0.0263 + 0.115i)16-s + (0.656 + 0.0739i)18-s + (0.433 + 0.900i)23-s + (0.730 + 0.730i)24-s + (0.623 + 0.781i)25-s + (−1.18 + 0.133i)26-s + (0.623 − 0.781i)27-s + ⋯
L(s)  = 1  + (−0.623 + 0.218i)2-s + (−0.222 + 0.974i)3-s + (−0.440 + 0.351i)4-s + (−0.0739 − 0.656i)6-s + (0.549 − 0.874i)8-s + (−0.900 − 0.433i)9-s + (−0.244 − 0.507i)12-s + (1.75 + 0.400i)13-s + (−0.0263 + 0.115i)16-s + (0.656 + 0.0739i)18-s + (0.433 + 0.900i)23-s + (0.730 + 0.730i)24-s + (0.623 + 0.781i)25-s + (−1.18 + 0.133i)26-s + (0.623 − 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.582 - 0.813i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.582 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6608047277\)
\(L(\frac12)\) \(\approx\) \(0.6608047277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 + (0.781 - 0.623i)T \)
good2 \( 1 + (0.623 - 0.218i)T + (0.781 - 0.623i)T^{2} \)
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 \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.974 + 0.222i)T^{2} \)
31 \( 1 + (-0.467 - 1.33i)T + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (1.19 + 1.19i)T + iT^{2} \)
43 \( 1 + (-0.781 - 0.623i)T^{2} \)
47 \( 1 + (0.189 - 0.119i)T + (0.433 - 0.900i)T^{2} \)
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 \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.0739 - 0.211i)T + (-0.781 - 0.623i)T^{2} \)
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 \( 1 + (-0.974 - 0.222i)T^{2} \)
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

−9.423112649453319307423668454034, −8.760265737016378500744732307246, −8.541050567189164320692299645774, −7.33598187938981573123460535977, −6.57049550754796243011753075599, −5.52713301894173496222323081820, −4.78038041865830258930273798060, −3.67606476733671884588125123749, −3.37524746362396714922390886457, −1.33062390745457513506420992777, 0.72271285857536692083285843453, 1.70772850594626610419998659872, 2.86376255901029625772121901024, 4.18691182927930136032648250833, 5.21545054033721427498903497312, 6.08438157840394592894830421999, 6.59352624529282676711032289106, 7.87685838180777490889288688514, 8.263713946379212325580767363223, 8.908708739790561717232703247066

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