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Properties

Label 2-1029-1029.827-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.417 - 0.908i$
Analytic cond. $0.513537$
Root an. cond. $0.716615$
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  + (−0.672 + 0.740i)3-s + (−0.981 − 0.191i)4-s + (0.718 + 0.695i)7-s + (−0.0960 − 0.995i)9-s + (0.801 − 0.598i)12-s + (0.395 + 1.07i)13-s + (0.926 + 0.375i)16-s − 1.89·19-s + (−0.997 + 0.0640i)21-s + (−0.345 + 0.938i)25-s + (0.801 + 0.598i)27-s + (−0.572 − 0.820i)28-s + (−1.22 + 1.53i)31-s + (−0.0960 + 0.995i)36-s + (0.783 + 1.76i)37-s + ⋯
L(s)  = 1  + (−0.672 + 0.740i)3-s + (−0.981 − 0.191i)4-s + (0.718 + 0.695i)7-s + (−0.0960 − 0.995i)9-s + (0.801 − 0.598i)12-s + (0.395 + 1.07i)13-s + (0.926 + 0.375i)16-s − 1.89·19-s + (−0.997 + 0.0640i)21-s + (−0.345 + 0.938i)25-s + (0.801 + 0.598i)27-s + (−0.572 − 0.820i)28-s + (−1.22 + 1.53i)31-s + (−0.0960 + 0.995i)36-s + (0.783 + 1.76i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5633577318\)
\(L(\frac12)\) \(\approx\) \(0.5633577318\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.672 - 0.740i)T \)
7 \( 1 + (-0.718 - 0.695i)T \)
good2 \( 1 + (0.981 + 0.191i)T^{2} \)
5 \( 1 + (0.345 - 0.938i)T^{2} \)
11 \( 1 + (-0.404 + 0.914i)T^{2} \)
13 \( 1 + (-0.395 - 1.07i)T + (-0.761 + 0.648i)T^{2} \)
17 \( 1 + (0.672 - 0.740i)T^{2} \)
19 \( 1 + 1.89T + T^{2} \)
23 \( 1 + (0.997 - 0.0640i)T^{2} \)
29 \( 1 + (0.997 + 0.0640i)T^{2} \)
31 \( 1 + (1.22 - 1.53i)T + (-0.222 - 0.974i)T^{2} \)
37 \( 1 + (-0.783 - 1.76i)T + (-0.672 + 0.740i)T^{2} \)
41 \( 1 + (0.949 - 0.315i)T^{2} \)
43 \( 1 + (0.476 + 1.60i)T + (-0.838 + 0.545i)T^{2} \)
47 \( 1 + (-0.967 + 0.253i)T^{2} \)
53 \( 1 + (0.572 - 0.820i)T^{2} \)
59 \( 1 + (0.949 + 0.315i)T^{2} \)
61 \( 1 + (-0.704 - 0.599i)T + (0.159 + 0.987i)T^{2} \)
67 \( 1 + (0.838 + 1.05i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.997 - 0.0640i)T^{2} \)
73 \( 1 + (-0.802 - 0.103i)T + (0.967 + 0.253i)T^{2} \)
79 \( 1 + (-1.37 - 0.660i)T + (0.623 + 0.781i)T^{2} \)
83 \( 1 + (-0.0320 + 0.999i)T^{2} \)
89 \( 1 + (0.462 - 0.886i)T^{2} \)
97 \( 1 + (-0.646 + 0.810i)T + (-0.222 - 0.974i)T^{2} \)
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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

−10.47076566197863441904601661692, −9.468231756460585710148962829623, −8.853341261711624685784907985887, −8.335270036918769040341081546773, −6.82307048658620486447162804275, −5.92929630128641587838180274522, −5.08966372651630827011460758154, −4.44785042137463979364005191836, −3.57324250306016659430691796945, −1.71466522952843232832211733397, 0.60863851960978421161463314355, 2.16346825822048426106161793184, 3.88499408373646752776207927611, 4.61750895710917151215273018192, 5.61018124160101621294781679506, 6.36716035164124421144458484849, 7.69551803388609583147232366724, 7.962050823270140146902743415223, 8.864315212529813220572897189878, 10.05435275426802339627736874792

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