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Properties

Label 2-1029-1029.869-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.840 - 0.541i$
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  + (−0.949 + 0.315i)3-s + (0.284 + 0.958i)4-s + (0.404 − 0.914i)7-s + (0.801 − 0.598i)9-s + (−0.572 − 0.820i)12-s + (1.91 − 0.503i)13-s + (−0.838 + 0.545i)16-s − 0.925·19-s + (−0.0960 + 0.995i)21-s + (0.967 + 0.253i)25-s + (−0.572 + 0.820i)27-s + (0.991 + 0.127i)28-s + (−0.126 − 0.554i)31-s + (0.801 + 0.598i)36-s + (0.295 + 1.83i)37-s + ⋯
L(s)  = 1  + (−0.949 + 0.315i)3-s + (0.284 + 0.958i)4-s + (0.404 − 0.914i)7-s + (0.801 − 0.598i)9-s + (−0.572 − 0.820i)12-s + (1.91 − 0.503i)13-s + (−0.838 + 0.545i)16-s − 0.925·19-s + (−0.0960 + 0.995i)21-s + (0.967 + 0.253i)25-s + (−0.572 + 0.820i)27-s + (0.991 + 0.127i)28-s + (−0.126 − 0.554i)31-s + (0.801 + 0.598i)36-s + (0.295 + 1.83i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.840 - 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9075417276\)
\(L(\frac12)\) \(\approx\) \(0.9075417276\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.949 - 0.315i)T \)
7 \( 1 + (-0.404 + 0.914i)T \)
good2 \( 1 + (-0.284 - 0.958i)T^{2} \)
5 \( 1 + (-0.967 - 0.253i)T^{2} \)
11 \( 1 + (-0.159 + 0.987i)T^{2} \)
13 \( 1 + (-1.91 + 0.503i)T + (0.871 - 0.490i)T^{2} \)
17 \( 1 + (0.949 - 0.315i)T^{2} \)
19 \( 1 + 0.925T + T^{2} \)
23 \( 1 + (0.0960 - 0.995i)T^{2} \)
29 \( 1 + (0.0960 + 0.995i)T^{2} \)
31 \( 1 + (0.126 + 0.554i)T + (-0.900 + 0.433i)T^{2} \)
37 \( 1 + (-0.295 - 1.83i)T + (-0.949 + 0.315i)T^{2} \)
41 \( 1 + (0.462 - 0.886i)T^{2} \)
43 \( 1 + (-0.525 - 1.42i)T + (-0.761 + 0.648i)T^{2} \)
47 \( 1 + (-0.926 - 0.375i)T^{2} \)
53 \( 1 + (-0.991 + 0.127i)T^{2} \)
59 \( 1 + (0.462 + 0.886i)T^{2} \)
61 \( 1 + (1.73 + 0.979i)T + (0.518 + 0.855i)T^{2} \)
67 \( 1 + (-0.422 + 1.85i)T + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.0960 - 0.995i)T^{2} \)
73 \( 1 + (0.313 - 0.0610i)T + (0.926 - 0.375i)T^{2} \)
79 \( 1 + (-1.08 - 1.36i)T + (-0.222 + 0.974i)T^{2} \)
83 \( 1 + (0.672 - 0.740i)T^{2} \)
89 \( 1 + (0.997 + 0.0640i)T^{2} \)
97 \( 1 + (0.0142 + 0.0624i)T + (-0.900 + 0.433i)T^{2} \)
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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

−10.67706853591500701043842535630, −9.430019511431127222872759162352, −8.376633707366184270539958721791, −7.78429249728294020534082959718, −6.64015134479591087675597579269, −6.21978017852351939483271038759, −4.83268933259618177356179697817, −4.05611433916706813866380900645, −3.23486197236470165542311482894, −1.33214184398805171970727920065, 1.27192834227639833586122395069, 2.26571819185469666451604989001, 4.09614945002712555205303066574, 5.12916318346731269567319900846, 5.93731431483871233496563639672, 6.35047242689191912607072438830, 7.31437590590633976824264011040, 8.633211846399758767989753689352, 9.115522204371862290354945113186, 10.50046238679797236564995411972

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