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Properties

Label 2-1029-147.134-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.914 - 0.404i$
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  + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)9-s + (0.900 + 0.433i)12-s + (0.277 + 1.21i)13-s + (0.623 − 0.781i)16-s + 0.445·19-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + (0.777 − 0.974i)39-s + (−0.277 + 0.347i)43-s − 48-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)9-s + (0.900 + 0.433i)12-s + (0.277 + 1.21i)13-s + (0.623 − 0.781i)16-s + 0.445·19-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)27-s + 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + (0.777 − 0.974i)39-s + (−0.277 + 0.347i)43-s − 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (932, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6529500851\)
\(L(\frac12)\) \(\approx\) \(0.6529500851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 - 0.445T + T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
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

−10.07317001936303045368065322300, −9.350979187623249525491940098020, −8.412374285018872769828762023299, −7.74321371292651084956022114672, −6.83260584459847423028986271651, −5.99071185751828297217412834147, −4.95072191335117094799114364588, −4.20046650870546801882749849997, −2.84626510279271662201747653688, −1.29779746923318993790539983109, 0.812785595516100100091452172450, 2.99837921584220364386510713262, 4.12465783019176481295097495596, 4.83180574446624923812420442823, 5.73855154183666145385855836925, 6.29518953325310664158918275347, 7.78464927709586585157879340979, 8.586848360309102503877419824834, 9.445935855000848874005791794298, 10.15239677864294787149096548384

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