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Properties

Label 2-1029-1029.8-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.770 - 0.637i$
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  + (−0.997 + 0.0640i)3-s + (0.967 + 0.253i)4-s + (0.518 + 0.855i)7-s + (0.991 − 0.127i)9-s + (−0.981 − 0.191i)12-s + (−0.476 − 0.310i)13-s + (0.871 + 0.490i)16-s − 0.192·19-s + (−0.572 − 0.820i)21-s + (−0.838 + 0.545i)25-s + (−0.981 + 0.191i)27-s + (0.284 + 0.958i)28-s + (1.20 + 1.51i)31-s + (0.991 + 0.127i)36-s + (−0.0488 − 1.52i)37-s + ⋯
L(s)  = 1  + (−0.997 + 0.0640i)3-s + (0.967 + 0.253i)4-s + (0.518 + 0.855i)7-s + (0.991 − 0.127i)9-s + (−0.981 − 0.191i)12-s + (−0.476 − 0.310i)13-s + (0.871 + 0.490i)16-s − 0.192·19-s + (−0.572 − 0.820i)21-s + (−0.838 + 0.545i)25-s + (−0.981 + 0.191i)27-s + (0.284 + 0.958i)28-s + (1.20 + 1.51i)31-s + (0.991 + 0.127i)36-s + (−0.0488 − 1.52i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9874295897\)
\(L(\frac12)\) \(\approx\) \(0.9874295897\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 - 0.0640i)T \)
7 \( 1 + (-0.518 - 0.855i)T \)
good2 \( 1 + (-0.967 - 0.253i)T^{2} \)
5 \( 1 + (0.838 - 0.545i)T^{2} \)
11 \( 1 + (-0.0320 + 0.999i)T^{2} \)
13 \( 1 + (0.476 + 0.310i)T + (0.404 + 0.914i)T^{2} \)
17 \( 1 + (0.997 - 0.0640i)T^{2} \)
19 \( 1 + 0.192T + T^{2} \)
23 \( 1 + (0.572 + 0.820i)T^{2} \)
29 \( 1 + (0.572 - 0.820i)T^{2} \)
31 \( 1 + (-1.20 - 1.51i)T + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (0.0488 + 1.52i)T + (-0.997 + 0.0640i)T^{2} \)
41 \( 1 + (0.0960 - 0.995i)T^{2} \)
43 \( 1 + (-1.33 + 0.539i)T + (0.718 - 0.695i)T^{2} \)
47 \( 1 + (0.761 + 0.648i)T^{2} \)
53 \( 1 + (-0.284 + 0.958i)T^{2} \)
59 \( 1 + (0.0960 + 0.995i)T^{2} \)
61 \( 1 + (-0.648 + 1.46i)T + (-0.672 - 0.740i)T^{2} \)
67 \( 1 + (1.24 - 1.56i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.572 + 0.820i)T^{2} \)
73 \( 1 + (0.0221 + 0.0601i)T + (-0.761 + 0.648i)T^{2} \)
79 \( 1 + (0.729 - 0.351i)T + (0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.462 + 0.886i)T^{2} \)
89 \( 1 + (-0.801 - 0.598i)T^{2} \)
97 \( 1 + (1.18 + 1.48i)T + (-0.222 + 0.974i)T^{2} \)
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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

−10.48765783601832327747635640492, −9.566570053937140955584031258382, −8.469388619100429491787656286114, −7.55062437593024992576052205254, −6.84126775359339534431737814855, −5.86140526167518329477340814310, −5.33174553658038276915958128637, −4.15252456510347815823462561936, −2.77031336674837876815356062320, −1.64325947211480441931171250469, 1.18147235018351113543939202734, 2.41691963759490042427266933143, 4.05803191902375905137702290828, 4.86126389043034079830830773502, 5.97287754743905638980999592390, 6.56365092918411960584141499951, 7.45701498791095478826797695956, 8.004334272606512990815591718017, 9.611823142019962170454440863394, 10.26553848161655595203305661074

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