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Properties

Label 2-1044-348.11-c0-0-1
Degree $2$
Conductor $1044$
Sign $-0.0859 + 0.996i$
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  + (0.330 + 0.943i)2-s + (−0.781 + 0.623i)4-s + (−1.52 − 0.734i)5-s + (−0.846 − 0.532i)8-s + (0.189 − 1.68i)10-s + (−1.52 − 0.347i)13-s + (0.222 − 0.974i)16-s + (−1.27 − 1.27i)17-s + (1.65 − 0.376i)20-s + (1.16 + 1.46i)25-s + (−0.175 − 1.55i)26-s + (−0.846 + 0.532i)29-s + (0.993 − 0.111i)32-s + (0.781 − 1.62i)34-s + (−0.119 + 0.189i)37-s + ⋯
L(s)  = 1  + (0.330 + 0.943i)2-s + (−0.781 + 0.623i)4-s + (−1.52 − 0.734i)5-s + (−0.846 − 0.532i)8-s + (0.189 − 1.68i)10-s + (−1.52 − 0.347i)13-s + (0.222 − 0.974i)16-s + (−1.27 − 1.27i)17-s + (1.65 − 0.376i)20-s + (1.16 + 1.46i)25-s + (−0.175 − 1.55i)26-s + (−0.846 + 0.532i)29-s + (0.993 − 0.111i)32-s + (0.781 − 1.62i)34-s + (−0.119 + 0.189i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0859 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0859 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.0859 + 0.996i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -0.0859 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2180473991\)
\(L(\frac12)\) \(\approx\) \(0.2180473991\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.330 - 0.943i)T \)
3 \( 1 \)
29 \( 1 + (0.846 - 0.532i)T \)
good5 \( 1 + (1.52 + 0.734i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.433 + 0.900i)T^{2} \)
13 \( 1 + (1.52 + 0.347i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (1.27 + 1.27i)T + iT^{2} \)
19 \( 1 + (-0.974 - 0.222i)T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (0.119 - 0.189i)T + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (-0.881 + 0.881i)T - iT^{2} \)
43 \( 1 + (0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.433 - 0.900i)T^{2} \)
53 \( 1 + (0.819 - 1.70i)T + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.0739 + 0.656i)T + (-0.974 + 0.222i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.78 + 0.623i)T + (0.781 + 0.623i)T^{2} \)
79 \( 1 + (-0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.643 - 1.84i)T + (-0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.222 - 1.97i)T + (-0.974 - 0.222i)T^{2} \)
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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

−9.437228395669835274752981064927, −8.977240494534598015801766690068, −7.981336540745818007730472027393, −7.43218586428814060697247488839, −6.82025609043650201658804689989, −5.35310056685504797787925893248, −4.73303383147633686402863505257, −4.03264830953805641741280466414, −2.83845820144671411037121710213, −0.17336867823329707730079317783, 2.12098470751756962051153584907, 3.14610019006567999140520522807, 4.15157338526803810413659694315, 4.63468539780962839558748414795, 6.05818977473521786131067000846, 7.06866526254507944675012765868, 7.894455688210125195547686603149, 8.760608468997348172948506115973, 9.738312434850925429771703658183, 10.55226459188550829416077699516

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