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Properties

Label 2-2044-2044.583-c0-0-0
Degree $2$
Conductor $2044$
Sign $0.997 - 0.0633i$
Analytic cond. $1.02008$
Root an. cond. $1.00999$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + i·7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.866 + 1.5i)11-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + 1.73·22-s + (−0.5 − 0.866i)25-s + (0.866 − 0.499i)28-s + (0.866 + 1.5i)31-s + (0.499 + 0.866i)32-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + i·7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.866 + 1.5i)11-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + 1.73·22-s + (−0.5 − 0.866i)25-s + (0.866 − 0.499i)28-s + (0.866 + 1.5i)31-s + (0.499 + 0.866i)32-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2044\)    =    \(2^{2} \cdot 7 \cdot 73\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(1.02008\)
Root analytic conductor: \(1.00999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2044} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2044,\ (\ :0),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295429410\)
\(L(\frac12)\) \(\approx\) \(1.295429410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - iT \)
73 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.493259229758408624431635846040, −8.736186597876574530673224116521, −7.982924314714466117883229746302, −6.73206007883253603553778850780, −6.05185087506616243258780030134, −4.97265061027901728251086434982, −4.64068524534257356137968703322, −3.39068695418547287569667014367, −2.38366615053297958656091882455, −1.71189999203795434980643288138, 0.822019970316140745236324668854, 2.92979020868731777894831579977, 3.78371012065945921495425871305, 4.24610892611390007363834493887, 5.68202336715234161137313837479, 6.03722543304363522380354553816, 6.90419409331468698359732180155, 7.61737697655501021014666066585, 8.487507990204391112921181395260, 9.084240475643674830074958922061

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