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Properties

Label 2-1044-116.63-c0-0-1
Degree $2$
Conductor $1044$
Sign $-0.303 + 0.952i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.541 − 0.678i)5-s + (−0.433 − 0.900i)8-s + (−0.846 − 0.193i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s − 1.24i·17-s + (−0.781 + 0.376i)20-s + (0.0549 − 0.240i)25-s + (0.433 − 0.0990i)26-s + (−0.433 + 0.900i)29-s + (−0.974 + 0.222i)32-s + (−0.777 − 0.974i)34-s + (0.846 + 1.75i)37-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.541 − 0.678i)5-s + (−0.433 − 0.900i)8-s + (−0.846 − 0.193i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s − 1.24i·17-s + (−0.781 + 0.376i)20-s + (0.0549 − 0.240i)25-s + (0.433 − 0.0990i)26-s + (−0.433 + 0.900i)29-s + (−0.974 + 0.222i)32-s + (−0.777 − 0.974i)34-s + (0.846 + 1.75i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.391236466\)
\(L(\frac12)\) \(\approx\) \(1.391236466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
3 \( 1 \)
29 \( 1 + (0.433 - 0.900i)T \)
good5 \( 1 + (0.541 + 0.678i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + 1.24iT - T^{2} \)
19 \( 1 + (-0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + 0.445iT - T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.974 - 1.22i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.40 + 1.12i)T + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (1.90 + 0.433i)T + (0.900 + 0.433i)T^{2} \)
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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

−9.940396220619285416121618278724, −9.199021113253846205215128684703, −8.339218211456007897016359057936, −7.23280050998631828408166535382, −6.34523308789971447531409146425, −5.24686892800962495987629772753, −4.60566827277671703259722127427, −3.66827021781147670672752313776, −2.59816251088589322282275009436, −1.10568362167479118874559339972, 2.27103505064598543055921434528, 3.53929802397891245558781547877, 4.04593060294455467576847797433, 5.32402672743495995299822395157, 6.15533481173252123410161316676, 6.91357624630833798684375177824, 7.79550598518188098872168054364, 8.345885342291876982465085359197, 9.436316526211819999369550415241, 10.63481554842160288485671549975

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