↑

Properties

Label 2-2001-2001.827-c0-0-1
Degree $2$
Conductor $2001$
Sign $0.347 - 0.937i$
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  + (0.222 + 1.97i)2-s + (−0.900 + 0.433i)3-s + (−2.87 + 0.656i)4-s + (−1.05 − 1.68i)6-s + (−1.27 − 3.65i)8-s + (0.623 − 0.781i)9-s + (2.30 − 1.83i)12-s + (−0.541 − 1.12i)13-s + (4.28 − 2.06i)16-s + (1.68 + 1.05i)18-s + (0.781 − 0.623i)23-s + (2.74 + 2.74i)24-s + (−0.222 − 0.974i)25-s + (2.09 − 1.31i)26-s + (−0.222 + 0.974i)27-s + ⋯
L(s)  = 1  + (0.222 + 1.97i)2-s + (−0.900 + 0.433i)3-s + (−2.87 + 0.656i)4-s + (−1.05 − 1.68i)6-s + (−1.27 − 3.65i)8-s + (0.623 − 0.781i)9-s + (2.30 − 1.83i)12-s + (−0.541 − 1.12i)13-s + (4.28 − 2.06i)16-s + (1.68 + 1.05i)18-s + (0.781 − 0.623i)23-s + (2.74 + 2.74i)24-s + (−0.222 − 0.974i)25-s + (2.09 − 1.31i)26-s + (−0.222 + 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.347 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5429295523\)
\(L(\frac12)\) \(\approx\) \(0.5429295523\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (-0.974 + 0.222i)T \)
good2 \( 1 + (-0.222 - 1.97i)T + (-0.974 + 0.222i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.781 + 0.623i)T^{2} \)
13 \( 1 + (0.541 + 1.12i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (1.40 - 0.158i)T + (0.974 - 0.222i)T^{2} \)
37 \( 1 + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (-1.00 - 0.351i)T + (0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + 0.445iT - T^{2} \)
61 \( 1 + (-0.433 + 0.900i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.05 + 0.119i)T + (0.974 + 0.222i)T^{2} \)
79 \( 1 + (0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (-0.433 - 0.900i)T^{2} \)
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

−9.254157532534492483556824369667, −8.530053523989609341612826006760, −7.72248291523525614181818396958, −6.94872157432104715391583867015, −6.36923405583838764858688129167, −5.49498799922210521031911374354, −5.05680275366472951691003303421, −4.25289431442849711188036126884, −3.29902373282357123616293954307, −0.46996291595628832652183201979, 1.27214741778403983367917596258, 2.04307029592682950993425709034, 3.18656055333057495945095168921, 4.20657730192278652477564379577, 4.96174960141199558182487938788, 5.56420854417074812399600787579, 6.73721873637553580073372873612, 7.76966597232715584235514153070, 8.859484827552639979738547935728, 9.487717257160239124450602885162

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