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Properties

Label 2-1044-1044.115-c0-0-0
Degree $2$
Conductor $1044$
Sign $-0.939 - 0.342i$
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  + (−0.5 + 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.499 − 0.866i)4-s + (−0.766 − 1.32i)5-s + (−0.939 − 0.342i)6-s + 0.999·8-s + (−0.939 + 0.342i)9-s + 1.53·10-s + (−0.766 + 1.32i)11-s + (0.766 − 0.642i)12-s + (0.939 + 1.62i)13-s + (1.17 − 0.984i)15-s + (−0.5 + 0.866i)16-s + (0.173 − 0.984i)18-s − 19-s + (−0.766 + 1.32i)20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.499 − 0.866i)4-s + (−0.766 − 1.32i)5-s + (−0.939 − 0.342i)6-s + 0.999·8-s + (−0.939 + 0.342i)9-s + 1.53·10-s + (−0.766 + 1.32i)11-s + (0.766 − 0.642i)12-s + (0.939 + 1.62i)13-s + (1.17 − 0.984i)15-s + (−0.5 + 0.866i)16-s + (0.173 − 0.984i)18-s − 19-s + (−0.766 + 1.32i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5206464494\)
\(L(\frac12)\) \(\approx\) \(0.5206464494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.23157678500607374223276911969, −9.388160980541064884531084975606, −8.718015628097226427944642302995, −8.354352999078194812061819758071, −7.30255608105837126416616441529, −6.30828199869579939622315326805, −4.95763922782785877442128826249, −4.70597847499171473981940107216, −3.86040748798263200210489207223, −1.75577401852371495279267461184, 0.56256932018251615924808017026, 2.45448054967584165898054479962, 3.09483128624849822638341808496, 3.85192811972893609074713234502, 5.70983137500984356066362104899, 6.49879849830326165353642427106, 7.73277581407621195255971019418, 7.989784586500443811315926624255, 8.639941261418912613675001999989, 10.02134946189661047841741145962

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