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Properties

Label 2-2044-2044.291-c0-0-1
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: $\chi_{2044} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2044,\ (\ :0),\ -0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9672079411\)
\(L(\frac12)\) \(\approx\) \(0.9672079411\)
\(L(1)\) 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

−9.327151527887792790001913187879, −8.947233756869108468647624907393, −8.029899934018987113597319868410, −7.24739832046858634768268204744, −6.60126987398262790096392481743, −5.52430144497344044638573151010, −5.25193305311067187187993552522, −4.11131941769775269611499841931, −3.14625876255394550909666513227, −2.16667374999513358154361896928, 0.55530706657110669692631933902, 2.10746567227129172603189829300, 3.03914992001668896080278639209, 3.88246306198453949217229640465, 4.79416406919865032826339211456, 5.63201295775174403501281275076, 6.25156989055431966382682895854, 7.56579569467137942712789811449, 8.208311600750909530973086667388, 8.995058225755487943241248056195

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