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Properties

Label 2-1029-147.11-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.0434 + 0.999i$
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  + (−0.988 + 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.955 − 0.294i)9-s + (0.0747 + 0.997i)12-s + (−0.277 − 1.21i)13-s + (−0.988 − 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (0.900 − 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.134 − 1.79i)37-s + (0.455 + 1.16i)39-s + (−0.277 + 0.347i)43-s + 0.999·48-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.955 − 0.294i)9-s + (0.0747 + 0.997i)12-s + (−0.277 − 1.21i)13-s + (−0.988 − 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (0.900 − 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.134 − 1.79i)37-s + (0.455 + 1.16i)39-s + (−0.277 + 0.347i)43-s + 0.999·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0434 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0434 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.0434 + 0.999i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.0434 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6719669046\)
\(L(\frac12)\) \(\approx\) \(0.6719669046\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.988 - 0.149i)T \)
7 \( 1 \)
good2 \( 1 + (-0.0747 + 0.997i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
79 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−9.914363126508631061329429082679, −9.596697516362718844040107937922, −8.180053457758333693130705472062, −7.27584823894648976497076982624, −6.22643050299288355965122117062, −5.70720278963695703247105299143, −4.93582195642617589152624198617, −3.90798508218781703799458124185, −2.24976659757639213257908347869, −0.72629210333105241839107935149, 1.75156317040646624121906754612, 3.18612379787838643818092983716, 4.35395177572868319988591433312, 5.02419548136428009721815165633, 6.33642319891838762922884936474, 6.93416687810977633319096877358, 7.67734276494445089857634703513, 8.678549467132518953664434876158, 9.535671400754801822113764356716, 10.47518104424875528965449279365

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