↑

Properties

Label 2-2001-2001.620-c0-0-4
Degree $2$
Conductor $2001$
Sign $-0.117 - 0.993i$
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  + (0.500 + 1.43i)2-s + (0.955 − 0.294i)3-s + (−1.01 + 0.809i)4-s + (0.900 + 1.21i)6-s + (−0.382 − 0.240i)8-s + (0.826 − 0.563i)9-s + (−0.731 + 1.07i)12-s + (0.145 + 0.0332i)13-s + (−0.136 + 0.597i)16-s + (1.21 + 0.900i)18-s + (−0.433 − 0.900i)23-s + (−0.436 − 0.116i)24-s + (0.623 + 0.781i)25-s + (0.0253 + 0.225i)26-s + (0.623 − 0.781i)27-s + ⋯
L(s)  = 1  + (0.500 + 1.43i)2-s + (0.955 − 0.294i)3-s + (−1.01 + 0.809i)4-s + (0.900 + 1.21i)6-s + (−0.382 − 0.240i)8-s + (0.826 − 0.563i)9-s + (−0.731 + 1.07i)12-s + (0.145 + 0.0332i)13-s + (−0.136 + 0.597i)16-s + (1.21 + 0.900i)18-s + (−0.433 − 0.900i)23-s + (−0.436 − 0.116i)24-s + (0.623 + 0.781i)25-s + (0.0253 + 0.225i)26-s + (0.623 − 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.117 - 0.993i$
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: \((2,\ 2001,\ (\ :0),\ -0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.085627347\)
\(L(\frac12)\) \(\approx\) \(2.085627347\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 + 0.294i)T \)
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 \( 1 + (0.974 + 0.222i)T^{2} \)
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

−9.219504305068013795673770723728, −8.363587328674148115182109340206, −8.018535315543953503518885521982, −7.08212643135279605374218308506, −6.62087860552018012339193420297, −5.74374701516961313995774824322, −4.76254252664900855908989605708, −4.00658326497134886431990884450, −3.00968072097131036704871116275, −1.72143489761037131323511587508, 1.45365314213183018153841147944, 2.35320469836463540503668349565, 3.23094730540402999340392806298, 3.89187480691116393684264467318, 4.67722719510831442053343709612, 5.57309033284444666799418765423, 6.93922988761389969672249006831, 7.70573795810221583867576403822, 8.695946928578710601530396189346, 9.270891230145568319129169882278

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