↑

Properties

Label 2-2040-2040.1589-c0-0-1
Degree $2$
Conductor $2040$
Sign $0.997 - 0.0758i$
Analytic cond. $1.01809$
Root an. cond. $1.00900$
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  + (−0.923 + 0.382i)2-s + (−0.923 − 0.382i)3-s + (0.707 − 0.707i)4-s + (0.382 − 0.923i)5-s + 6-s + (−0.382 + 0.923i)8-s + (0.707 + 0.707i)9-s + i·10-s + (−0.923 + 0.382i)12-s + (−0.707 + 0.707i)15-s − i·16-s + (0.923 + 0.382i)17-s + (−0.923 − 0.382i)18-s + (1 + i)19-s + (−0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (−0.923 − 0.382i)3-s + (0.707 − 0.707i)4-s + (0.382 − 0.923i)5-s + 6-s + (−0.382 + 0.923i)8-s + (0.707 + 0.707i)9-s + i·10-s + (−0.923 + 0.382i)12-s + (−0.707 + 0.707i)15-s − i·16-s + (0.923 + 0.382i)17-s + (−0.923 − 0.382i)18-s + (1 + i)19-s + (−0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(1.01809\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :0),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6367297579\)
\(L(\frac12)\) \(\approx\) \(0.6367297579\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
good7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (-0.765 - 1.84i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 0.765iT - T^{2} \)
53 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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

−9.348754642230605880773487958138, −8.551979007393516308899067729346, −7.65081719504778226589074607041, −7.20801484440224264744960130394, −6.10540294763639099556049630134, −5.45489412018738535369433195175, −5.07596425788274392012259521260, −3.48412772011012033563231061583, −1.74464045302598124404421968050, −1.14573485593915830108450432637, 0.879669988089516664534386409687, 2.42625035424587306405309619557, 3.25373429057987455377137517868, 4.35546264552307470296462566892, 5.52712174888817175213784377428, 6.29106240702754268233435584379, 7.09235225180451774114402228841, 7.55761827609348539796696687378, 8.809297545374420374669421557419, 9.600221091806931117569441366892

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