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Properties

Label 2-1029-147.137-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.843 - 0.537i$
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  + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (0.365 − 0.930i)9-s + (−0.955 + 0.294i)12-s + (1.12 − 1.40i)13-s + (0.826 + 0.563i)16-s + (0.623 + 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.385i)31-s + (0.623 − 0.781i)36-s + (−0.425 + 0.131i)37-s + (−0.134 + 1.79i)39-s + (−1.12 + 0.541i)43-s − 48-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (0.365 − 0.930i)9-s + (−0.955 + 0.294i)12-s + (1.12 − 1.40i)13-s + (0.826 + 0.563i)16-s + (0.623 + 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.385i)31-s + (0.623 − 0.781i)36-s + (−0.425 + 0.131i)37-s + (−0.134 + 1.79i)39-s + (−1.12 + 0.541i)43-s − 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.032896345\)
\(L(\frac12)\) \(\approx\) \(1.032896345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (-0.425 + 0.131i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.440 - 0.0663i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 + 1.24T + T^{2} \)
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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

−10.36002456506838622344720825619, −9.706374881295096549834128601237, −8.403926167801806886714392587823, −7.72831336095240453608191637964, −6.66425127890899405156496659119, −5.90042563437518018562570525279, −5.30180423088534990690818817211, −3.81063196938956558839737491474, −3.19428908761890692069643952985, −1.45787612644303473304765656070, 1.36123770231437638260681195598, 2.34339192911859290235671729217, 3.86247151117826676854689636441, 5.11240339851570512045339263357, 5.97393966691238016481118973597, 6.70480513300943739209860660107, 7.22045777244940328627959157667, 8.252955798464592660186318856318, 9.340842046937341904073386809063, 10.28399086005880459912660822482

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