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Properties

Label 2-1029-1029.113-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.715 + 0.698i$
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  + (−0.838 + 0.545i)3-s + (−0.672 + 0.740i)4-s + (−0.981 + 0.191i)7-s + (0.404 − 0.914i)9-s + (0.159 − 0.987i)12-s + (−0.479 + 0.919i)13-s + (−0.0960 − 0.995i)16-s − 1.52·19-s + (0.718 − 0.695i)21-s + (−0.462 − 0.886i)25-s + (0.159 + 0.987i)27-s + (0.518 − 0.855i)28-s + (0.299 − 1.31i)31-s + (0.404 + 0.914i)36-s + (−0.567 − 1.91i)37-s + ⋯
L(s)  = 1  + (−0.838 + 0.545i)3-s + (−0.672 + 0.740i)4-s + (−0.981 + 0.191i)7-s + (0.404 − 0.914i)9-s + (0.159 − 0.987i)12-s + (−0.479 + 0.919i)13-s + (−0.0960 − 0.995i)16-s − 1.52·19-s + (0.718 − 0.695i)21-s + (−0.462 − 0.886i)25-s + (0.159 + 0.987i)27-s + (0.518 − 0.855i)28-s + (0.299 − 1.31i)31-s + (0.404 + 0.914i)36-s + (−0.567 − 1.91i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.715 + 0.698i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05164541610\)
\(L(\frac12)\) \(\approx\) \(0.05164541610\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.838 - 0.545i)T \)
7 \( 1 + (0.981 - 0.191i)T \)
good2 \( 1 + (0.672 - 0.740i)T^{2} \)
5 \( 1 + (0.462 + 0.886i)T^{2} \)
11 \( 1 + (-0.284 + 0.958i)T^{2} \)
13 \( 1 + (0.479 - 0.919i)T + (-0.572 - 0.820i)T^{2} \)
17 \( 1 + (0.838 - 0.545i)T^{2} \)
19 \( 1 + 1.52T + T^{2} \)
23 \( 1 + (-0.718 + 0.695i)T^{2} \)
29 \( 1 + (-0.718 - 0.695i)T^{2} \)
31 \( 1 + (-0.299 + 1.31i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.567 + 1.91i)T + (-0.838 + 0.545i)T^{2} \)
41 \( 1 + (0.761 - 0.648i)T^{2} \)
43 \( 1 + (1.52 - 0.505i)T + (0.801 - 0.598i)T^{2} \)
47 \( 1 + (0.997 + 0.0640i)T^{2} \)
53 \( 1 + (-0.518 - 0.855i)T^{2} \)
59 \( 1 + (0.761 + 0.648i)T^{2} \)
61 \( 1 + (0.996 - 1.42i)T + (-0.345 - 0.938i)T^{2} \)
67 \( 1 + (-0.372 - 1.63i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (-0.718 + 0.695i)T^{2} \)
73 \( 1 + (-0.0182 - 0.568i)T + (-0.997 + 0.0640i)T^{2} \)
79 \( 1 + (0.713 - 0.894i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.926 - 0.375i)T^{2} \)
89 \( 1 + (-0.871 + 0.490i)T^{2} \)
97 \( 1 + (0.430 - 1.88i)T + (-0.900 - 0.433i)T^{2} \)
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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

−10.51973042600087601122925574652, −9.750629787465027831736804140350, −9.179224369970187856499376714427, −8.360768235299445731270110730559, −7.13348781609562158449957339958, −6.41463507466856547129758713309, −5.49709076791419823187768939751, −4.28696661098267849440558701113, −3.94820711134729945748295090299, −2.52404152561837180039978948668, 0.05280546650264401915961247649, 1.65561512784634521492150492032, 3.28045695457937385178626121333, 4.63206452921348648247089597319, 5.32319429706758578131105319103, 6.29033937290877756170190248164, 6.76965749553689294046061334621, 7.951947452493629768170428753708, 8.828431885708660794250871592358, 9.953488208324858023397887208324

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